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arXiv:1704.03761v1 [cs.IT] 12 Apr 2017
From ds-bounds for cyclic codes to true distance
for abelian codes.
J.J. Bernal, M. Guerreiro∗and J. J. Sim´on †‡
April 13, 2017
Abstract
In this paper we develop a technique to extend any bound for the
minimum distance of cyclic codes constructed from its defining sets (ds-
bounds) to abelian (or multivariate) codes through the notion of B-apparent
distance. We use this technique to improve the searching for new bounds
for the minimum distance of abelian codes. We also study conditions for
an abelian code to verify that its B-apparent distance reaches its (true)
minimum distance. Then we construct some tables of such codes as an
application.
Keywords: Abelian code, bounds for minimum distance, true minimum dis-
tance, algorithm.
1 Introduction
The study of abelian codes is an important topic in Coding Theory, having
an extensive literature, because they have good algebraic properties that allow
one to construct good codes with efficient encoding and decoding algorithms.
More precisely, regarding decoding, the two most known general techniques are
permutation decoding [2] and the so-called locator decoding [6] that uses the
Berlekamp-Massey algorithm [21] (see also [16]).
Even though the mentioned decoding methods require to know the mini-
mum distance, or a bound for it, there are not much literature or studies on its
computation and properties, or it does exist only for specific families of abelian
codes (see [6]). Concerning BCH bound, in [10], P. Camion introduces an ex-
tension from cyclic to abelian codes which is computed through the apparent
∗M. Guerreiro is with Departamento de Matem´atica, Universidade Federal de Vi¸cosa,
36570-900 Vi¸cosa-MG, Brazil. Supported by CNPq-Brazil, Processo PDE 233497/2014-5.
E-mail: marines@ufv.br
†J. J. Bernal and J. J. Sim´on are with the Departamento de Matem´aticas, Universidad de
Murcia, 30100 Murcia, Spain. Partially supported by MINECO pro ject MTM2016-77445-P
and Fundaci´on S´eneca of Murcia. E-mail: {josejoaquin.bernal, jsimon}@um.es
‡This work was partially presented at the ISITA 2016, Monterey, CA, USA.
1
distance of such codes. Since then there have appeared some papers improving
the original computation and giving a notion of multivariate BCH bound and
codes (see, for example, [3, 20]).
These advances lead us to some natural questions about the extension to the
multivariate case of all generalizations and improvements of the BCH bound
known for cyclic codes; specifically, those bounds on the minimum distance for
cyclic codes which are defined from defining sets.
There are dozens of papers on this topic regarding approaches from matrix
methods ([5, 25]) through split codes techniques ([12, 15]) until arriving at
the most classical generalizations based on computations over the defining set,
as the Hartmann-Tzeng (HT) bound [13], the Ross (R) bound [19] and the
improvements by Van Lint and Wilson, as the shifting bound (SB) [22].
Having so many references on the subject, it seems very necessary to find a
general method that allows one to extend any bound for the minimum distance
of cyclic codes based on the defining set to the multivariate case. This is our
first goal. We shall show a method to extend to the multivariate case any bound
of the mentioned type via associating an apparent distance to such bound.
The second target of our study is to improve the searching for new bounds
for abelian codes. At this point, we must honestly say that these searches may
only have interest for codes whose minimum distances are not known (in fact, if
one knows the minimum distance of a code one does not need a bound for it), so
that, our examples consider codes of lengths necessarily large. In our opinion,
long abelian codes are not so bad (see [1]) in terms of performance.
As we work with long codes, certainly it seems impossible at the moment
to compute their minimum distance, and so, it is natural to ask for conditions
on them to ensure that a founded bound is in fact the minimum distance. This
is the last goal of this paper. We found conditions for a bivariate abelian code
to reach the mentioned equality and we write these conditions in terms of its
defining set from the notion of composed polynomial matrices (CP-matrices,
for short). We comment the extension of this results to several variables. We
illustrate with some examples (of large codes) how this technique works.
In order to achieve our goals, we give in Section 3 a notion of defining set
bound (ds-bound) for the minimum distance of cyclic codes. In Section 4, we
revisit the relation between the weight of codewords of abelian codes and the
apparent distance of their discrete Fourier transforms. In Section 5, we use
this technique to define the apparent distance of an abelian code with respect
to a set of ds-bounds. In Section 6, we adapt a known algorithm given in [3]
(of linear complexity by Remark 23) to compute the B-apparent distance of
an abelian code. Finally, we study the abelian codes which verify the equality
between its BCH bound and its minimum distance. For two variables, we find
some sufficient conditions that are easy to extend to several variables.
2
2 Preliminaries
Let Fqbe a finite field with qelements, with qa power of a prime p,ribe
positive integers, for all i∈ {1,...,s}, and n=r1···rs. We denote by Zrithe
ring of integers modulo riand we shall always write its elements as canonical
representatives.
An abelian code of length nis an ideal in the algebra Fq(r1,...,rs) =
Fq[X1,...,Xs]/hXr1
1−1,...,Xrs
s−1iand throughout this work we assume
that this algebra is semisimple; that is, gcd(ri, q) = 1, for all i∈ {1,...,s}.
Abelian codes are also called multidimensional cyclic codes (see, for example,
[14]).
The codewords are identified with polynomials f(X1,...,Xs) in which, for
each monomial, the degree of the indeterminate Xkbelongs to Zrk. We denote
by Ithe set Zr1× · · · × Zrsand we write the elements f∈Fq(r1,...,rs) as
f=f(X1,...,Xs) = PaiXi, where i= (i1,...,is)∈Iand Xi=Xi1
1···Xis
s.
Given a polynomial f∈Fq[X1,...,Xs] we denote by fits image under the
canonical projection onto Fq(r1,...,rs).
For each i∈ {1,...,s}, we denote by Rri(resp. Uri) the set of all ri-th
roots of unity (resp. all ri-th primitive roots of unity) and define R=Qs
i=1 Rri
(U=Qs
i=1 Uri).
For f=f(X1,...,Xs)∈Fq[X1,...,Xs] and ¯α∈R, we write f( ¯α) =
f(α1,...,αs). For i= (i1,...,is)∈I, we write ¯αi= (αi1
1,...,αis
s).
It is a known fact that every abelian code Cin Fq(r1,...,rs) is totally
determined by its root set or set of zeros, namely
Z(C) = {¯α∈R|f(¯α) = 0,for all f∈C}.
The set of non zeros is denoted by Z(C) = R\Z(C). For a fixed ¯α∈U, the
code Cis determined by its defining set, with respect to ¯α, which is defined
as
D¯α(C) = i∈I|f(¯αi) = 0,for all f∈C.
Given an element a= (a1,...,as)∈I, we shall define its q-orbit mod-
ulo (r1,...,rs) as Q(a) = a1·qi,...,as·qi∈I|i∈N. In the case of
a semisimple algebra, it is known that any defining set D¯α(C) is a disjoint
union of q-orbits modulo (r1,...,rs). Conversely, every union of q-orbits mod-
ulo (r1,...,rs) determines an abelian code (an ideal) in Fq(r1,...,rs) (see, for
example, [3] for details). We note that the notions of root set and defining
set also apply to polynomials. Moreover, if Cis the ideal generated by the
polynomial fin Fq(r1,...,rn), then D¯α(C) = D¯α(f).
We recall that the notion of defining set also applies to cyclic codes. For
s= 1 and r1=n, a q-orbit is called a q-cyclotomic coset of a positive integer
bmodulo nand it is the set Cq(b) = {b·qi∈Zn|i∈N}.
Throughout this paper, we fix the notation L|Fqfor an extension field con-
taining Uri, for all i∈ {1,...,s}. The discrete Fourier transform of a
3
polynomial f∈Fq(r1,...,rs)with respect to ¯α∈U(also called Mattson-
Solomon polynomial in [20]) is the polynomial ϕ¯α,f (X) = Pj∈If(¯αj)Xj∈
L(r1,...,rs).It is known that the discrete Fourier transform may be viewed
as an isomorphism of algebras ϕ¯α:L(r1,...,rs)−→ (L|I|, ⋆),where the mul-
tiplication “⋆” in L|I|is defined coordinatewise. Thus, we may see ϕ¯α,f as a
vector in L|I|or as a polynomial in L(r1,...,rs) (see [10, Section 2.2]). The
inverse of the discrete Fourier transform is ϕ−1
¯α,g(X) = 1
r1r2···rsPj∈Ig(¯α−j)Xj.
3 Defining set bounds for cyclic codes
In this section we deal with cyclic codes; that is, r1=n. By P(Zn) we denote
the power set of Zn. We take an arbitrary α∈Un.
Definition 1 Adefining set bound (or ds-bound, for short) for the mini-
mum distance of cyclic codes is a family of relations δ={δn}n∈Nsuch that, for
each n∈N,δn⊆ P(Zn)×Nand it satisfies the following conditions:
1. If Cis a cyclic code in F(n)such that N⊆ Dα(C), then 1≤a≤d(C),
for all (N, a)∈δn.
2. If N⊆Mare subsets of Znthen (N , a)∈δnimplies (M , a)∈δn.
3. For all N∈ P (Zn),(N , 1) ∈δn.
From now on, sometimes we write simply δto denote a ds-bound or any of
its elements independently on the length nof the code. It will be clear from the
context which one is being used.
Remarks 2 (1) For example, the BCH bound states that, for any cyclic code
in Fq(n) which in its set of zeros has a string of t−1 consecutive powers of some
α∈Un, the minimum distance of the code is at least t[17, Theorem 7.8].
Now, define δ⊂ P(Zn)×Nas follows: for any a≥2, (N, a)∈δif and only if
there exist i0, i1,...,ia−2in Nwhich are consecutive integers modulo n. Then
the BCH bound says that δis a ds-bound for any cyclic code (we only have to
state Condition 3 as a convention; so that (∅,1) ∈δBCH ).
(2) It is easy to check that all extensions of the BCH bound, all new bounds
from the defining set of a cyclic code as in [5, 13, 18, 19, 25] and the new bounds
and improvements arising from Corollary 1, Theorem 5 and results in Section 4
and Section 5 in [22], also verify Definition 1.
In general, for any bound for the minimum distance of a cyclic code, say
b, we denote the corresponding ds-bound by δb. In order to relate the idea of
ds-bound with the Camion’s apparent distance, which will be defined later, we
consider the following family of maps.
Definition 3 Let δbe a ds-bound for the minimum distance of cyclic codes.
The optimal ds-bound associated to δis the family δ={δn}n∈Nof maps
δn:P(Zn)−→ Ndefined as δn(N) = max{b∈N|(N, b)∈δn}.
4
The following result is immediate.
Lemma 4 Let δbe a ds-bound for the minimum distance of cyclic codes. Then,
for each n∈Z:
1. If Cis a cyclic code in F(n)such that N⊆ Dα(C), then 1≤δn(N)≤
d(C).
2. If N⊆M⊆Zn, then δn(N)≤δn(M).
As we noted above, we may omit the index of the map ¯
δn, because it will be
clear from the context for which value it is being taken.
4 Apparent distance of matrices
We begin this section recalling the notion and notation of a hypermatrix that
will be used hereby, as it is described in [3]. For any i∈I=Zr1× · · · × Zrs,
we write its k-th coordinate as i(k). A hypermatrix with entries in a field
Sindexed by I(or an I-hypermatrix over S)is an s-dimensional I-array,
denoted by M= (ai)i∈I, with ai∈S[23]. The set of indices, the dimension
and the ground field will be omitted if they are clear from the context. For
s= 2, Mis a matrix and when s= 1, Mis a vector. We write M= 0 when all
its entries are 0 and M6= 0, otherwise. As usual, a hypercolumn is defined
as HM(j, k) = {ai∈M|i(j) = k}, with 1 ≤j≤sand 0 ≤k < rj, where
ai∈Mmeans that aiis an entry of M. A hypercolumn can be seen as an
(s−1)-dimensional hypermatrix. In the case s= 2, we refer to hypercolumns
as rows or columns and, when s= 1, we say entries.
For any I-hypermatrix Mwith entries in a field, we define the support of M
as the set supp(M) = {i∈I|ai6= 0}. Its complement with respect to Iwill
be denoted by D(M). When D(M) (or supp(M)) is an union of q-orbits we say
that Mis a q-orbits hypermatrix. Let D⊆I. The hypermatrix afforded
by Dis defined as M= (ai)i∈I, where ai= 1 if i6∈ Dand ai= 0 otherwise;
it will be denoted by M=M(D). Note that if Dis union of q-orbits then
M(D) is a q-orbits hypermatrix. To define and compute the apparent distance
of an abelian code we will use the hypermatrix afforded by its defining set, with
respect to ¯α∈U.
We define a partial ordering on the set {M(D)|Dis union of q-orbits in I}
as follows:
M(D)≤M(D′)⇔supp (M(D)) ⊆supp (M(D′)) .(1)
Clearly, this condition is equivalent to D′⊆D.
We begin with the apparent distance of a vector in Ln.
Definition 5 Let δbe a ds-bound for the minimum distance of cyclic codes
and v∈Lna vector. The apparent distance of vwith respect to δ(or
δ-apparent distance of v, for short), denoted by δ∗(v), is defined as
5
1. If v= 0, then δ∗(v) = 0.
2. If v6= 0, then δ∗(v) = δ(Zn\supp(v)).
From now on we denote by Ba set of ds-bounds which are used to proceed
a computation of the apparent distances of matrices, hypermatrices or abelian
codes.
Definition 6 Let v∈Ln. The apparent distance of vwith respect to B
denoted by ∆B(v), is:
1. If v= 0, then ∆B(v) = 0.
2. If v6= 0, then ∆B(v) = max{δ∗(v)|δ∈B}.
Remarks 7 The following properties arise straightforward from the definition
above, for any v∈Ln.
1. If v6= 0 then ∆B(v)≥1.
2. If supp(v)⊆supp(w)then ∆B(v)≥∆B(w).
Proposition 8 Let f∈L(n)and vbe the vector of its coefficients. Fix any
α∈Un. Then ∆B(v)≤ω(ϕ−1
α,f ) = |Z(f)|.
Proof. Set N=Zn\supp(v) and let Cbe the abelian code generated by ϕ−1
α,f
in L(n). Then d(C)≤ω(ϕ−1
α,f ). By properties of the discrete Fourier transform,
we have N=Dα(ϕ−1
α,f ) = Dα(C) hence, by Lemma 4 and the definition of
apparent distance, ∆B(v)≤d(C). This gives the desired inequality. The last
equality is obviuos.
The notion of apparent distance appeared for the first time in [10] and orig-
inally it was defined for polynomials. Its computation reflects a bound of the
nonzeros (in the sense given in the preliminaries) of a multivariate polynomial.
The aim of the apparent distance was to extend the notion of BCH bound, from
cyclic to abelian codes as we will comment in the following paragraphs. The
first algorithm for its computation was made in terms of coefficients of poly-
nomials. Later, in [20], R. E. Sabin gave an algorithm in terms of matrices.
The notion of strong apparent distance, that appeared in [3], is a slight but
powerful modification of the original one, defined for multivariate polynomials
and hypermatrices, and it is the predecessor of the current apparent distance
defined with respect to a list of ds-bounds.
Remark 9 To identify notations from previous works with the ones used here,
given a polynomial f∈Fq(r1,...,rs), if we denote by M(f) its hypermatrix of
coefficients (in the obvious sense), then the strong apparent distance of fin [3]
is sd∗(f) = ∆δBCH (M(f)); that is, B={δBC H }, together with the convention
that δBC H (∅) = 1.
6
Now let us show how the notion of apparent distance for abelian codes works
as the BCH bound for cyclic codes. All results in the following corollary are
proved in [10] and [20].
Corollary 10 Let Cbe a cyclic code in Fq(n)and α∈Un. Then
1. If g , e ∈Care the generating polynomial and the idempotent generator of
C, respectively, then ∆B[M(ϕα,g )] = ∆B[M(ϕα,e)] ≤∆B[M(ϕα,c )], for
all c∈C.
2. If c∈Cis a codeword with ϕα,c =f∈L(n), then ω(c)≥∆B(M(f)) and
consequently
3. ∆B[M(ϕα,g)] = ∆B[M(ϕα,e )] = min {∆B[M(ϕα,c)] |c∈C} ≤ d(C).
The number on the left of the last inequality is known as the apparent
distance of the cyclic code Cwith respect to the set Band α∈Unor
the B-apparent distance of Cwith respect to α∈Un.
Proof. (1) comes from the fact that, for all c∈C, we have supp (M(ϕα,c)) ⊆
supp (M(ϕα,g)) = supp (M(ϕα,e)), together with Remark 7. (2) is Proposi-
tion 8. (3) is immediate from (1) and (2).
Now we shall define the apparent distance of matrices and hypermatrices
with respect to a set Bof ds-bounds.
Definition 11 Let Mbe an s-dimensional I-hypermatrix over a field L. The
apparent distance of Mwith respect to B, denoted by ∆B(M), is defined
as follows:
1. ∆B(0) = 0 and, for s= 1, Definition 6 applies.
2. For s= 2 and a nonzero matrix M, note that HM(1, i)is the i-th row
and HM(2, j)is the j-th column of M. Define the row support of Mas
supp1(M) = {i∈ {0,...r1−1} | HM(1, i)6= 0}and the column support
of Mas supp2(M) = {k∈ {0,...r2−1} | HM(2, k)6= 0}.
Then put
ω1(M) = max{δ(Zr1\supp1(M)) |δ∈B},
ǫ1(M) = max{∆B(HM(1, j)) |j∈supp1(M)}
and set ∆1(M) = ω1(M)·ǫ1(M).
Analogously, we compute the apparent distance ∆2(M)for the other vari-
able and finally we define the apparent distance of Mwith respect to
Bby
∆B(M) = max{∆1(M),∆2(M)}.
7
3. For s > 2, proceed as follows: suppose that one knows how to compute the
apparent distance ∆B(N), for all non zero hypermatrices Nof dimension
s−1. Then first compute the “hypermatrix support” of M6= 0 with respect
to the j-th hypercolumn, that is,
suppj(M) = {i∈ {0,...rj−1} | HM(j, i)6= 0}.
Now put
ωj(M) = max{δ(Zrj\suppj(M)) |δ∈B},
ǫj(M) = max{∆B(HM(j, k)) |k∈suppj(M)}
and set ∆j(M) = ωj(M)·ǫj(M).
Finally, define the apparent distance of Mwith respect to B(or the
B-apparent distance) as:
∆B(M) = max {∆j(M)|j∈ {1,...,s}} .
As we have already commented in Remark 9, by taking B={δBCH }, ∆B(M)
is the strong apparent distance in [3].
Now, as in the case of cyclic codes, we relate the apparent distance to the
weight of codewords. For each multivariate polynomial f=Pi∈IaiXi, consider
the hypermatrix of the coefficients of f, denoted by M(f) = (ai)i∈I. For any
j∈ {0,...,s}, if we write f=Prj−1
k=0 fj,kXk
j, where fj,k =fj,k (Xj) and Xj=
X1···Xj−1·Xj+1 ···Xs, then M(fj,k) = HM(j, k). This means that “fixed”
the variable Xjin f, for each power kof Xj, the coefficient fj,k is a polynomial
in Xj, and HM(j, k) is the hypermatrix obtained from its coefficients. Now we
extend Proposition 8 to several variables.
Theorem 12 Let f∈L(r1,...,rs)and M=M(f)be the hypermatrix of its
coefficients. Fix any ¯α∈U. Then ∆B(M(f)) ≤ωϕ−1
α,f =|Z(f)|.
Proof. For f= 0, the result is obvious. Consider f6= 0. The case n= 1
is Proposition 8. We prove the theorem for matrices; that is, for s= 2. The
general case follows directly by induction.
Set M=M(f)6= 0, ¯α= (α1, α2) and write f=Pr2−1
k=0 f2,kXk
2. Then
HM(2, k) is the vector of coefficients of f2,k. Clearly, supp2(M) = {k∈
{0,...,r2−1} | f2,k 6= 0}and, for any k∈supp2(M), we have ∆B(HM(2, k)) ≥
1. Then ωϕ−1
α1,f2,k ≥1, by Proposition 8.
Now, for each fixed k∈Zr2, by the definition of discrete Fourier transform,
|Z(f2,k)|=ω(ϕ−1
α1,f2,k ), hence, if k∈supp2(M) there exists t∈Zr1(at least
one) such that αt
1is a non zero of f2,k .
Set g(X2) = f(αt
1, X2) = Pr2−1
k=0 f2,k(αt
1)Xk
2and note that it is a non zero
polynomial. Let v(g) be the vector of coefficients of g(X2). Then supp (v(g)) ⊆
supp2(M) and so ∆B(v(g)) ≥max{¯
δ(Zr2\supp2(M)) |δ∈B}=ω2(M).
8
As |Z(g)|=ω(ϕ−1
α2,g )≥∆B(v(g)) then, for any k∈Zr2,
|Z(f)| ≥ |Z(g)| · |Z(f2,k)| ≥ ∆B(v(g)) ·ω(ϕ−1
α1,f2,k )≥ω2(M)·∆B(HM(2, k)) .
Finally, as in the univariate case, it is clear that |Z(f)|=ω(ϕ−1
¯α,f ). The exten-
sion to more variables is clear and this completes the proof.
Example 13 Set n= 96 = 4 ×24 and q= 5. Fix α1∈U4and α2∈U24 and
consider the 5-orbits matrix Mafforded by D=Q(0,0) ∪Q(0,1) ∪Q(0,2) ∪
Q(0,3)∪Q(0,6)∪Q(0,7)∪Q(0,9) ∪Q(1,1) ∪Q(1,2)∪Q(1,3)∪Q(2,1)∪Q(2,2)∪
Q(3,6).Choose B={δB S , δBC H }, where δBS is the Betti-Sala bound in [5].
One may check that supp1(M) = {0,1,2,3}and then ω1(M) = 1. On the
other hand, supp2(M) = Z24 so ω2(M) = 1. Now ∆B(HM(1,0)) = 8, by using
δBS (see [5, Example 4.2]) which is the maximum of the values of the two bounds
considered, hence ǫ1(M) = 8. It is clear that ǫ2(M) = 4, so that ∆1(M) = 8
and ∆2(M) = 4. Hence ∆{δBS,δBC H }(M) = 8.
The computation of the apparent distance in several variables is a natural
extension of that of one variable, and, moreover, the relationship between ap-
parent distance and weight of a codeword is essentially the same in any case.
However, condition (2) of Remark 7(2) does not necessarily hold in two or more
variables and so we cannot extend directly the results of Corollary 10. Let us
show the situation in the following example.
Example 14 Let Mbe the 2-orbits matrix of order 5 ×7 such that supp(M) =
Q(0,0) ∪Q(1,0) ∪Q(1,3) and Nbe the 2-orbits matrix such that supp(N) =
Q(1,0) ∪Q(1,3). Then N < M , however, one may check that ∆δBCH (N) = 6
and ∆δBCH (M) = 7. So Remark 7(2) does not hold in this case.
As we will see in the next section, the notion of B-apparent distance of an
abelian code with respect to some roots of unity will be a natural extension
of that of cyclic codes, while its computation will be an interesting problem to
solve.
5 The B-apparent distance of an abelian code
The following definition changes a little the usual way to present the notions of
apparent distance from [10] and the strong apparent distance from [3] (see also
[20]). We recall that Bdenotes a set of ds-bounds, which are used to proceed a
concrete computation of the apparent distances.
Definition 15 Let Cbe an abelian code in Fq(r1,...,rs).
1) The apparent distance of Cwith respect to ¯α∈Uand B(or the
(B,¯α)-apparent distance) is
∆B,α(C) = min{∆B(M(ϕ¯α,c)) |c∈C}.
9
2) The apparent distance of Cwith respect to Bis
∆B(C) = max{∆B,¯α(C)|¯α∈U}.
The following result is a consequence of Theorem 12.
Corollary 16 For any abelian code Cin Fq(r1,...,rs)and any Bas above,
∆B(C)≤d(C).
Proof. Let g∈Csuch that ω(g) = d(C). By Theorem 12, ∆B(M(ϕ¯α,g)) ≤
ω(g), for any ¯α∈U. From this, the result follows directly.
It is certain that to compute the apparent distance for each element of a
code in order to obtain its apparent distance can be as hard work as to compute
the minimum distance of such a code. Thus, to improve the efficiency of the
computation the following result tells us that we may restrict our attention to
the idempotents of the code. It also allows us to reformulate Definition 15 as it
is presented in the previously mentioned papers.
Proposition 17 Let Cbe an abelian code in Fq(r1,...,rs). The apparent dis-
tance of Cwith respect to ¯α∈Uand Bverifies the equality
∆B,¯α(C) = min{∆B(M(ϕ¯α,e )) |e2=e∈C}.
Proof. Consider any c∈C. Since Fq(r1,...,rs) is semisimple, then there
exists an idempotent e∈Csuch that the ideals generated by cand ein
Fq(r1,...,rs) coincide; that is, hci=hei, and so D¯α(c) = D¯α(e). This means
that supp (M(ϕ¯α,c )) = supp (M(ϕ¯α,e)). Note that the computation of the ap-
parent distance is based on the fact that the entries (of the matrices) are zero
or not; that is, once an entry is non zero, its specific value is irrelevant. From
this fact, it is easy to see that ∆B(M(ϕ¯α,c)) = ∆B(M(ϕ¯α,e)) and so we get the
desired equality.
Let e∈Fq(r1,...,rs) be an idempotent and Ebe the ideal generated by
e. Then ϕ¯α,e ⋆ ϕ¯α,e =ϕ¯α,e, for any ¯α∈Uand thus, if ϕ¯α,e =Pi∈IaiXi, we
have ai∈ {1,0} ⊆ Fqand ai= 0 if and only if i∈ D¯α(E). Hence M(ϕ¯α,e) =
M(D¯α(E)). Conversely, let Mbe a hypermatrix afforded by a set Dwhich is a
union of q-orbits. We know that Ddetermines a unique ideal Cin Fq(r1,...,rs)
such that D¯α(C) = D. Let e∈Cbe its generating idempotent. Clearly,
M(ϕ¯α,e) = M(D).
Now let Cbe an abelian code, ¯α∈Uand let Mbe the hypermatrix
afforded by D¯α(C). For any q-orbits hypermatrix P≤M[see the order-
ing (1)] there exists a unique idempotent e′∈Csuch that P=M(ϕ¯α,e′)
and, for any codeword f∈C, there is a unique idempotent e(f) such that
10
∆B(M(ϕ¯α,f )) = ∆BM(ϕ¯α,e(f)). Therefore,
min{∆B(P)|06=P≤M}=
min{∆B(M(ϕ¯α,e)) |06=e2=e∈C}= ∆B,¯α(C).
This fact drives us to give the following definition.
Definition 18 For a q-orbits hypermatrix M, its minimum B-apparent dis-
tance is
B−mad(M) = min{∆B(P)|06=P≤M}.
Finally, in the next theorem we set the relationship between the apparent
distance of an abelian code and the minimum apparent distance of hypermatri-
ces.
Theorem 19 Let Cbe an abelian code in Fq(r1,...,rs)and let ebe its gen-
erating idempotent. For any ¯α∈U, we have ∆B,¯α(C) = B−mad (M(ϕ¯α,e)).
Therefore, ∆B(C) = max{B−mad (M(ϕ¯α,e)) |¯α∈U}.
Proof. It follows directly from the preceding paragraphs.
6 Computing minimum apparent distance
In [3] it is presented an algorithm to find, for any abelian code, a list of matrices
(or hypermatrices in case of more than 2 variables) representing some of its
idempotents whose apparent distances based on the BCH bound (called the
strong apparent distance) go decreasing until the minimum value is reached.
It is a kind of “suitable idempotents chase through hypermatrices” [3, p. 2].
This algorithm is based on certain manipulations of the (q-orbits) hypermatrix
afforded by the defining set of the abelian code. It is not so hard to see that it
is possible to obtain an analogous algorithm in our case.
We reproduce here the result and the algorithm in the case of two variables
under our notation. Then we will use the mentioned algorithm to improve the
searching for new bounds for abelian codes.
Definition 20 With the notation of the previous sections, let Dbe a union of
q-orbits and M=M(D)the hypermatrix afforded by D. We say that HM(j, k)
is an involved hypercolumn (row or column for two variables) in the
computation of ∆B(M), if ∆B(HM(j, k)) = ǫj(M)and ∆j(M) = ∆B(M).
We denote the set of indices of involved hypercolumns by Ip(M). Note that
the involved hypercolumns are those which contribute in the computation of the
B-apparent distance.
The next result shows a sufficient condition to get at once the minimum
B-apparent distance of a hypermatrix.
11
Proposition 21 With the notation as above, let Dbe a union of q-orbits and
M=M(D)the hypermatrix afforded by D. If ∆B(HM(j, k)) = 1, for some
(j, k)∈Ip(M), then B−mad(M) = ∆B(M).
Proof. It is a modification of that in [3, Proposition 23] having in mind the
use of different ds-bounds.
Theorem 22 Let Qbe the set of all q-orbits modulo (r1, r2),µ∈ {1,...,|Q|−1}
and {Qj}µ
j=1 a subset of Q. Set D=∪µ
j=1Qjand M=M(D). Then there exist
two sequences: the first one is formed by nonzero q-orbits matrices, M=M0>
···> Ml6= 0 and the second one is formed by positive integers m0≥ · · · ≥ ml,
with l≤µand mi≤∆B(Mi), verifying the following property:
If Pis a q-orbits matrix such that 06=P≤M, then ∆B(P)≥mland if
∆B(P)< mi−1then P≤Mi, where 0< i ≤l.
Moreover, if l′∈ {0,...,l}is the first element satisfying ml′=mlthen
∆B(Ml′) = B−mad(M).
Proof. It follows the same lines of that in [3, Proposition 25] having in mind
the use of different ds-bounds.
Algorithm for matrices.
Set I=Zr1×Zr2. Consider the q-orbits matrix M= (mij )(i,j)∈Iand a set
Bof ds-bounds.
Step 1. Compute the apparent distance of Mwith respect to Band set
m0= ∆B(M).
Step 2.
a) If there exists (j, k)∈Ip(M) (see notation below Definition 20) such
that ∆B(HM(j, k)) = 1, then we finish giving the sequences M=M0
and m0= ∆B(M) (because of Proposition 21).
b) If ∆B(HM(j, k)) 6= 1, for all (k, b)∈Ip(M), we set
S=[
(k,b)∈Ip(M)
supp(HM(k, b))
and construct the q-orbits matrix M1= (aij )(i,j)∈Isuch that
aij =(0 if (i, j)∈ ∪{Q(k , b)|(k, b)∈S}
mij otherwise.
In other words, M1< M is the (q-orbits) matrix with maximum support
such that the involved rows and columns of Mare replaced by zero. One
may prove that if 0 6=P < M and ∆B(P)< m0then P≤M1.
Step 3.
12
a) If M1= 0, then we finish giving the sequences M=M0and m0=
∆B(M).
b) If M16= 0, we set m1= min{m0,∆B(M1)}, and we get the sequences
M=M0> M1and m0≥m1. Then, we go back to Step 1 with M1
in the place of Mand m1in the place of m0.
Remark 23 If the q-orbits matrix has µ q-orbits, the algorithm has at most µ
steps.
Example 24 We take the setting of Example 13 and consider the abelian code
Cwith D(C) = D. In this case, the matrix M=M(D(C)) is the same as
that in the mentioned example. Choose again B={δBCH , δB S }. This code has
dimF5(C) = 73 and ∆B(C) = 8.
As Ip(M) = {(1,0)}, the matrix M1has the first row all zero and the others
equal to the ones of M; that is D(M1) = (∪i∈Z24 Q(0, i)) ∪Q(1,1) ∪Q(1,2) ∪
Q(1,3) ∪Q(2,1) ∪Q(2,2) ∪Q(3,6).
Now supp1(M1) = {1,2,3},ω1(M1) = 2 and ǫ1(M1) = 4 and we also have
supp2(M1) = {0,...,23},ω2(M1) = 1 and ǫ2(M1) = 4. Hence ∆B(M1) = 8.
Here Ip(M1) = {(1,1)}and we get the matrix M2having its first and second
rows all zero and the others equal to the ones of Mand M1; that is, D(M2) =
(∪i∈Z24 Q(0, i)) ∪(∪i∈Z24 Q(1, i)) ∪Q(2,1) ∪Q(2,2) ∪Q(3,6).
Here supp1(M2) = {2,3},ω1(M2) = 3 and ǫ1(M2) = 3 and we also have
supp2(M2) = {0,...,23},ω2(M2) = 1 and ǫ2(M2) = 4. Hence ∆B(M2) = 9
and Ip(M2) = {(1,2)}and we get the matrix M3having its first, second and
third rows all zero and the others equal to the ones of M,M1and M2; that is,
D(M2) = (∪i∈Z24 Q(0, i)) ∪(∪i∈Z24 Q(1, i)) ∪(∪i∈Z24 Q(2, i)) ∪Q(3,6).
Finally, supp1(M3) = {3},ω1(M2) = 4 and ǫ1(M2) = 2; so we also have
supp2(M2) = Z24 \ {6},ω2(M3) = 2 and ǫ2(M3) = 4. Hence ∆B(M3) = 8 and
Ip(M3) = {(1,3)}and we get M4= 0. Therefore, ∆{δBCH ,δBS }(M) = 8.
The closest code to Cwe know is a (105,51,7) binary cyclic code in [12, Table
II]. The known bounds for linear codes with the same length and dimension are
between 10 and 15.
The reader may find some tables with examples of this kind in [4].
7 True minimum distance in abelian codes
In this section we study the problem of find abelian codes such that its apparent
distance or its multivariate BCH bound reaches its minimum distance. We
keep all the notation from the preceding sections. In [8, 9] it is presented a
characterization of cyclic and BCH codes whose apparent distance reaches their
minimum distance. Our aim is to extend those results for multivariate codes.
Theorem 25 Let Cbe an abelian code in Fq(r1,...,rs). The following condi-
tions are equivalent:
13
1. ∆B(C) = d(C).
2. There exist an element α∈Uand a codeword c∈Csuch that its image
under the discrete Fourier transform, g=ϕα,c, verifies:
(a) ∆B(M(g)) = ∆B,α(C) = min {∆B(M(ϕα,v)) |v∈C}.
(b) ∆B(M(g)) = Z(g).
Proof. [1 ⇒2] Let c∈Ca be codeword with ω(c) = d(C) and α∈Usuch
that ∆B,α(C) = ∆B(C). Set g=ϕα,c . Then
d(C) = ∆B(C) = ∆B,α(C)≤∆B(M(g)) ≤Z(g)=ω(c) = d(C),
where the first equality is given by hypothesis. Thus, the inequality becomes
equality.
[2 ⇒1] Suppose that there is a codeword c∈Csatisfying the hypotheses.
Then
d(C)≤ω(c) = Z(g)= ∆B(M(g)) = ∆B,α(C)≤∆B(C)≤d(C)
and, again, equalities hold.
Remarks 26 The conditions in statement (2) of Theorem 25 refers only to a
single element in U. This is an important reduction that will be very useful
later. On the other hand, we recall that if Mis the hypermatrix afforded by
Dα(C) then ∆B,α(C) = B−mad(M).
For a given abelian code, the problem of finding, if any, a codeword verifying
Condition (2) of Theorem 25 is in general difficult to solve. In the case that the
codeword is an idempotent we will be able to find it through the computation of
the minimum apparent distance. So far we only know an actual way to find the
desired idempotent codeword of the mentioned theorem and it is given in the
following result.
Proposition 27 Let Cbe a code in Fq(r1, r2),α∈Uand Mthe matrix afforded
by its defining set Dα(C). Let P≤Mbe a q-orbits matrix and e∈L(r1, r2)be
the idempotent such that that P=M(e). If Pverifies
1. ∆(P)B=B−mad(M)and
2. ∆(P)B=Z(e).
Then d(C) = B−mad(M) = ∆B(C).
14
Proof. Since P≤M, then ϕ−1
α,e ∈C, hence ω(ϕ−1
α,e)≥d(C)≥∆B(C). On the
other hand, by the hypothesis (2),
ω(ϕ−1
α,e) = ∆B(P) = B−mad(M)≤∆B(C)≤d(C).
Therefore, d(C) = B−mad(M) = ∆B(C).
So, for a given code Cwith afforded matrix M, if we want to know whether
d(C) = ∆B(C) by using Proposition 27, in the case that the requested code-
word is an idempotent, we have to analyze all (q-orbits) matrices P≤M. If
Dα(C)=tand such idempotent exists, we have to do at most 2tsteps. This is
a search on the set of idempotents of C. The reader may note that the original
computation of the apparent distance in [10] and [20] requires to compute the
apparent distance of exactly the same set of q-orbits matrices. This might be
an important reduction in some cases.
We wonder if it is possible to simplify the procedure to find a q-orbits matrix
P, as in Proposition 27, by analyzing the sequence of matrices in the algorithm
for the computation of the minimum apparent distance of the matrix Mafforded
by Dα(C); i.e. the computation of B−mad(M). The algorithm gives us an
interesting reduction.
In the algorithm for the computation of the strong apparent distance as in
Theorem 22, we consider the sequence of matrices
M=M0> M1>···> Mj0−1> Mj0>··· > Mℓ>0 (2)
and let j0be the first index such that ∆B(Mj0) = mℓ=B−mad(M). If
m0= ∆B(M) equals mℓ, then P=Mand we do not have any reduction.
However, if m0> mℓ, then ∆B(P) = mℓ< mj0−1which implies P≤Mj0,
hence we can start our search from Mj0; that is, we have to check only at most
2t−j0matrices in order to find the hypothetical matrix of Theorem 25
We wonder if the existence of a matrix P≤Msatisfying the conditions
of Proposition 27 implies the existence of a matrix in the sequence (2) also
satisfying those conditions. The answer is negative, as the following very simple
example shows.
Example 28 Set ∆ = ∆B, with B={δBC H }. There exists an abelian code C,
with matrix Mafforded by Dα(C)with respect to α∈Usuch that:
1. For every q-orbits matrix in the sequence M=M0>··· >0we have
∆(Mj)6=Z(ej), where ej∈L(r1, r2)is the idempotent that verifies
Mj=M(ej).
2. d(C) = ∆(C)
Proof. Set q= 2 and r1=r2= 7. Let Cbe the code such that Dα(C) =
Q(0,3) ∪Q(1,3) ∪Q(1,5) ∪Q(1,6) ∪Q(3,0) ∪(3,2) ∪Q(3,3) ∪Q(3,4) ∪Q(3,5) ∪
15
Q(3,6) with respect to α∈U. The matrix afforded by Dα(C) is
M=
1 1 1 0 1 0 0
1 1 1 0 1 0 0
1 1 1 0 1 0 0
0 1 0 0 0 0 0
1 1 1 0 1 0 0
0 0 0 0 1 0 0
0 0 1 0 0 0 0
.
Let a(X1) = (1 + X1)(1 + X2
1+X3
1), b(X2) = (1 + X2)(1 + X2
2+X3
2). If
e∈Cis the idempotent generator then ϕα,e (X1, X2) = a(X1)b(X2) + X3
1X2+
X6
1X2
2+X3
1X4
2. One may compute Z(ϕα,e )= 25, by using GAP.
On the other hand, computing B−mad(M), we obtain the chain M0>0
and ∆(M) = B−mad(M) = 9. Now consider the q-orbits matrix
P=M(ab) =
1 1 1 0 1 0 0
1 1 1 0 1 0 0
1 1 1 0 1 0 0
0 0 0 0 0 0 0
1 1 1 0 1 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
.
Note that Pdoes not belong to the sequence M0>0; however, as a|X7
1−1
and b|X7
2−1, then g1=ab satisfies the hypothesis of Proposition 34 which
means that Csatisfies the condition (2b) of Theorem 25. Now, as ∆(P) = 9
then condition (2a) of the same theorem is satisfied and thus d(C) = ∆(C).
As we have seen, although Proposition 27 gives us a sufficient condition, it
does not guarantee that we can find the desired codeword if it is not an idempo-
tent, not even by using the algorithm. Now, in order to construct codes Csatis-
fying d(C) = ∆B(C), we try to move forward into a different direction: we firstly
characterize those polynomials that verify (2b); that is, ∆B(M(g)) = |Z(g)|. In
the univariate case, those polynomials were characterized in [9]. Before to ex-
tend the results to multivariate polynomials, we need to put some restrictions
on the election of ds-bounds that we may use. In the case of polynomials in one
variable, one may see that the condition (2b) in Theorem 25 forces us to use
exclusively the BCH bound as, to the best of our knowledge, the computation
of |Z(g)|is only known to be obtained in terms of the degree of Xhg(viewed
as a polynomial); that is, a list of consecutive exponents of the monomials with
coefficient zero of the highest degrees.
In the reminder of this section, consider B={δBC H }and denote ∆ =
∆δBCH , for the sake of simplicity. Let us recall some facts from univariate
polynomials that will be used herein. For 0 6=g∈L(n), let mg= gcd(Xn−
1, Xhg), which does not depend on h∈N. As we pointed out in Remark 9 about
16
notation, sd∗= ∆{δBCH }= ∆. The proof of the following result is essentially
the same as [9, Proposition 1]; so we ommit it.
Proposition 29 Let g∈L(n). Then
∆(M(g)) = Z(g)if and only if Xhg|Xn−1, for some h∈N.
Consider g=g(X1, X2)∈L(r1, r2) and write M=M(g). In general, as
we have seen in Theorem 12, ∆(M(g)) ≤Z(g). We want to describe the
polynomials gin L(r1, r2) such that ∆(M(g)) = Z(g)so we assume that the
equality holds; moreover, we impose the following condition
∆1(M) = ∆2(M) = ∆(M) = Z(g).(3)
where, as in Definition 11, ∆i(M) = ωi(M)·ǫi(M), for i∈ {1,2}.
We also write, following the notation of paragraph prior to Theorem 12,
g= [g(X1)](X2) =
r2−1
X
k=0
g2,kXk
2, g = [g(X2)](X1) =
r1−1
X
k=0
g1,kXk
1.(4)
For all u∈L, the polynomials of the form g(u, X2) and g(X1, u) have the
obvious meaning. For j= 1,2, denote by Zj=πjZ(g)the projection of
Z(g) onto the j-th coordinate and we also write Zj=πj(Z(g)).
For each j∈ {1,2}, we set Mj={k∈ {0,...,ri−1} | (j, k )∈Ip(M)}; that
is, ∆ (M(gj,k)) = εj, for any k∈Mj. Note that Ip(M) = (1, M1)∪(2, M2), (see
Definition 20).
Now, for each k∈M1, define
D1,k =n(u, v)∈R|v∈Z(g1,k ) and u∈Z(g(X1, v))o
and analogously, for each k∈M2, define
D2,k =n(u, v)∈R|u∈Z(g2,k ) and v∈Z(g(u, X2))o.
So, if we set, for j∈ {1,2},Dj,k = min nZ(g(u, Xj))|u∈Z(gj,k )othen
Z(gj,k )·Dj,k≤ |Dj,k |.
We know that, for each k∈M1, it happens ε1(M) = ∆ (M(g1,k )) ≤
Z(g1,k )and for each v∈Z(g1,k), we have ω1(M)≤∆(M(g(X1, v))) and so
ω1(M)≤D1,k , hence ε1(M)ω1(M)≤ |D1,k |. However, by the condition (3),
ε1(M)ω1(M) = ∆1(M) = Z(g)and, by definition, |D1,k| ≤ Z(g). Therefore,
for all k∈M1,
Z(g)=ε1(M)ω1(M) = |D1,k|and so D1,k =Z(g).(5)
17
Analogously, D2,k =Z(g), for k∈M2.
In fact, for j∈ {1,2}and any k∈Mj, we have Z(gj,k )·Dj,k =|Dj,k |,
hence
ε1(M) = ∆ (M(g1,k)) = Z(g1,k)and ω1(M) = D1,k.(6)
Keeping in mind the equalities obtained in the previous paragraphs, we get
the following two results.
Lemma 30 Let g=g(X1, X2)∈L(r1, r2)be a polynomial such that M=M(g)
satisfies the condition (3). Then:
1. For each k∈M1,∆ (M(g1,k)) = Z(g1,k )=Z2and ∆(M(g(X1, v))) =
Z(g(X1, v)), for any v∈Z(g1,k ).
2. For each k∈M2,∆ (M(g2,k)) = Z(g2,k )=Z1and ∆(M(g(u, X2))) =
Z(g(u, X2)), for any u∈Z(g2,k ).
Proof. We prove (1.) as the the proof of (2.) is entirely analogous. As
we have already seen, if condition (3) is satisfied then (5) and (6) also hold; so
that, ε1(M) = ∆ (M(g1,k)) = Z(g1,k).
Once we have the first equality, if ω1(M) = ∆ (M(g(X1, v))) <Z(g(X1, v )),
for some v∈Z(g1,k), then it must happen ε1(M)ω1(M)<|D1,j |, a contradic-
tion. Finally, if v∈Z2, then there exists u∈Rr1such that (u, v)∈Z(g) = D1,k,
hence u∈Z(g1,k). This proves the other equalities of this lemma.
Proposition 31 Let g=g(X1, X2)∈L(r1, r2)be a polynomial such that M=
M(g)satisfies the condition (3). Then there exist a=a(X1)∈L(r1), b =
b(X2)∈L(r2)and F=F(X1, X2)∈L(r1, r2)such that g=abF and
1. Xh1
1a|(Xr1
1−1), for some h1∈Z1, with ∆(M(a)) = ε2(M).
2. Xh2
2b|(Xr2
2−1), for some h2∈Z2, with ∆(M(b)) = ε1(M).
Proof. By Lemma 30.1 and by Proposition 29, for each k∈M2, if we denote
mk= gcd (g2,k, X r1
1−1), then
∆(M(mk)) = Z(mk)=r1− |Z(mk)|=r1− |Z(g2,k)|=
=Z(g2,k )= ∆ (M(g2,k )) .
By definition, ∆ (M(g2,k)) = r1−deg(Xk′
1g2,k), for some k′∈N. As
∆(M(mk)) = ∆ (M(g2,k)) and, by [9, Lemma 2], ∆(M(mk)) = r1−deg mk,
then Xk′
1g2,k and mkare associated.
18
Now we claim that mk|g2,j, for all j∈ {0,...,r2−1}. Indeed, for a fixed
k∈M2, by (5), we have D2,k =Z(g) which implies Z(g2,j )⊆Z(g2,k) or rather
Z(g2,k)⊆Z(g2,j ). Hence, mk|g2,j , for all j∈ {0,...,r2−1}.
Denoting by g′
2,j =g2,j
mk, for all j∈ {0,...,r2−1}and a(X1) = mk, we may
write
g(X1)(X2) = a(X1)
r2−1
X
j=0
g′
2,j Xj
2,(7)
with ∆(M(a(X1))) = ε2(M).
Analogously, we get
g(X2)(X1) = b(X2)
r1−1
X
i=0
g′
1,iXi
1,(8)
with ∆(M(b(X2))) = ε1(M) and b(X2)g′
1,i =g1,i, for any i∈ {0,...,r1−1}.
It is important to note that 1 = gcd Xr1
1−1, g′
2,0,...,g′
2,r2−1and 1 =
gcd Xr2
2−1, g′
1,0,...,g′
1,r1−1.
Now by writing
f(X1, X2) =
r2−1
X
j=0
g′
2,j Xj
2and h(X1, X2) =
r1−1
X
i=0
g′
1,iXi
1,
we get g(X1, X2) = a(X1)f(X1, X2) = b(X2)h(X1, X2).
Recall that Z1=π1(Z(g)) and Z2=π2(Z(g)).
First note that if v∈Z2, then there exists u∈Z1such that (u, v)∈Z(g).
This implies (u, v)∈D1,k , for k∈M1, hence v∈Z(g2,k ) = Z(b). Therefore,
Z2⊆Z(b) and, by Lemma 30.1, Z2=Z(b).
Consider g(X1, v) = b(v)h(X1, v) = b(v)Pr1−1
i=0 g′
2,i(v)Xi
1, for any v∈Rr2.
If b(v)6= 0, then g(X1, v)6= 0, otherwise all g′
2,i would have a common zero,
which is not possible. Conversely, if g(X1, v)6= 0, then b(v)6= 0. This proves
that v∈Z(b) = Z2if and only if g(X1, v)6= 0.
Obviously g(X1, v) = 0 also implies b(v) = 0, hence v∈Z2, as Z(b)⊆Z2.
Now let us write
f(X1, X2) =
r1−1
X
i=0
f2,iXi
1and h(X1, X2) =
r2−1
X
j=0
h1,j Xj
2.
If v∈Z(b), then g(X1, v) = 0 and we have f(X1, v) = 0. Since a(X1)6= 0 we
must have f2,i(v) = 0, for all i∈ {0,...,r1−1}. Hence Z(b)⊂Z(fi) and b(X2)|
f2,i, for all i∈ {0,...,r1−1}. Now if f2,i(v) = 0, for all i∈ {0,...,r1−1},
then f(X1, v) = 0 and g(X1, v) = 0, which implies v∈Z(b), as we have seen
before. Hence b(X2) = gcd(Xr2
2−1, f2,i), for all i∈ {0,...,r1−1}. Therefore,
f(X1, X2) = b(X2)f′(X1, X2) and
g(X1, X2) = a(X1)b(X2)f′(X1, X2).(9)
19
Analogously, one may prove that a(X1)|h1,j, for all j∈ {0,...,r2−1}, and
get h(X1, X2) = a(X1)h′(X1, X2), hence
g(X1, X2) = a(X1)b(X2)h′(X1, X2).(10)
Finally, note that the decompositions g=abf ′and g=abh′from (9)
and (10), has been done in L[X1, X2], which is a domain, and so, we have
f′(X1, X2) = h′(X1, X2). By writing F(X1, X2) = f′(X1, X2) = h′(X1, X2), we
get
g(X1, X2) = a(X1)b(X2)F(X1, X2).
It is clear that the condition (3) plays an important role in all the previous
proofs. Recall that a polynomial g∈L(r1, r2), with coefficient matrix M=
M(g), satisfies such condition if
∆1(M) = ∆2(M) = ∆(M) = Z(g).
For those polynomials, we have obtained a factorization g=abF , which
describes them, where ∆(M(a)) = ε2(M) and ∆(M(b)) = ε1(M).
At this point, two questions arise for an abelian code satisfying Theorem 25
with a codeword image g=ϕα,c as in such theorem.
1. Is it always true that gsatisfies also condition (3)?
2. Suppose a polynomial g∈L(r1, r2) already satisfies condition (3) and so
we have a decomposition g=abF . What can we say about F? More
specifically, is it true that M(F) is a q-orbits matrix? And is it true that
∆(M(a)) = ∆(M(b)) = ∆(M(g))?
We shall answer all these questions in the following examples.
Example 32 There exists an abelian code Cgenerated by an idempotent e∈
Fq(r1, r2), with image g=ϕα,f , satisfying the following properties.
1. ∆1(M(g)) <∆2(M(g)) (so the condition (3) is not fully satisfied)
2. ∆(M(g)) = Z(g).
3. d(C) = ∆(C).
Proof. Set q= 2, r1= 5, r2= 9 and Cbe the code with D(C) = Q(1,3), a
minimal code with generator idempotent e(X1, X2) = X4
1X7
2+X3
1X8
2+X4
1X6
2+
X2
1X8
2+X3
1X6
2+X4
1X4
2+X3
1X5
2+X2
1X6
2+X1X7
2+X4
1X3
2+X2
1X5
2+X1X6
2+
X3
1X3
2+X4
1X2+X3
1X2
2+X2
1X3
2+X1X4
2+X4
1+X2
1X2
2+X1X3
2+X3
1+X2
1+
X1X2+X1. Using the program GAP, we computed d(C) = 24. One may
check that ϕα,e =g(X1, X2) = X1X3
2+X4
1X3
2+X2
1X6
2+X3
1X6
2. Some di-
rect computations yield ∆1(M(g)) = 18 and ∆(M(g)) = 24, so assertion (1)
20
of this example is satisfied. As ω(e) = 24 we also get assertion (2). Since
Cis minimal, we have, 24 = ∆(M(g)) = B−mad(M(g)). On the other hand
∆(M(g)) ≤∆(C)≤d(C) = 24. Thus d(C) = ∆(C) and we get assertion (3).
The previous example gives a negative answer to question 1. The following
example answers question 2.
Example 33 Under the same notation of Proposition 31, there exists an abelian
code Cgenerated by an idempotent e∈Fq(r1, r2), with image g=ϕα,e, such
that the following properties hold.
1. M(g)satisfies the condition (3) and then g=abF as in the mentioned
proposition (see paragraph prior to Example 32).
2. d(C) = ∆(C).
3. F(X1, X2)has at least two nonzero monomials and M(F)is not a q-orbits
matrix.
4. ∆(a) = ∆(b), but ∆(a)∆(b)6= ∆(M(g))
5. Z(g)6=Z1×Z2
Proof. Let q= 2, r1=r2= 5 and Cbe the code with D(C) = Q(1,1) ∪Q(1,3).
In this case, g(X1, X2) = X4
1X4
2+X3
1X4
2+X4
1X2
2+X3
1X3
2+X2
1X2
2+X1X3
2+
X2
1X2+X1X2and ϕ−1
α,g =X3
1X4
2+X4
1X2
2+X4
1X2+X3
1X2
2+X2
1X3
2+X1X4
2+
X1X3
2+X2
1X2, so that |Z(g)|= 8.
By using GAP, we computed d(C) = 8 and one may check that ∆1(M(g)) =
∆2(M(g)) = ∆(M(g)) = 8; so that (1.) and (2.) hold.
Let us factorize g. In the case
g(X1)(X2) = (X1+X2
1)X2+ (X2
1+X4
1)X2
2+ (X1+X3
1)X3
2+ (X3
1+X4
1)X4
2,
M2={1,4}and a=a(X1) = 1 + X1. Note that (1 + X1)X1is a common factor
in all summands of g(X1)(X2). On the other hand,
g(X2)(X1) = (X2+X3
2)X1+ (X2+X2
2)X2
1+ (X3
2+X4
2)X3
1+ (X2
2+X4
2)X4
1,
M1={2,3}and b=b(X2) = 1 + X2. Here (1 + X2)X2is a common factor in
all summands of g(X2)(X1). Moreover,
f(X2)(X1) = (X2+X3
2)X1+ (X2
2+X3
2)X2
1+ (X2
2+X4
2)X3
1
h(X1)(X2) = (X1+X2
1)X2+ (X1+X4
1)X2
2+ (X3
1+X4
1)X3
2and so
F(X1, X2) = X1X2+X1X2
2+X2
1X2
2+X3
1X2
2+X3
1X3
2.
This gives us (3.)
Now, one may easily check that ∆(M(a)) = ∆(M(b)) = 4, hence ∆(M(a)) ·
∆(M(b)) 6= ∆(M(g)). This gives us (4).
21
Finally, by using GAP we compute Z(g) = Q(1,3) ∪Q(1,4) and clearly
Z(g)6=Z1×Z2.
To finish our argumentation from Theorem 25 we prove that a polynomial
that satisfies condition (3), and so factorizes g=abF with F(X1, X2) a mono-
mial in L(r1, r2), verifies that its image under the discrete Fourier transform
satisfy condition (2) of the mentioned theorem.
Proposition 34 Suppose g∈L(r1, r2)is such that g(X1, X2) = a(X1)b(X2),
where aand bsatisfy Proposition 29. Set M=M(g). Then
1. Z(g) = Z1×Z2
2. ∆(M) = ∆(M(a)) ·∆(M(b)) = Z(g).
3. ∆1(M) = ∆2(M) = ∆(M) = Z(g)(the condition (3)).
4. ∆(M(a)) = ε2(M) = ω1(M)and
5. ∆(M(b)) = ε1(M) = ω2(M).
Proof. Assertion (1) and the equality ∆(M(a)) ·∆(M(b)) = Z(g)come
directly from the decomposition of gtogether with the hypothesis that aand b
satisfy Proposition 29.
Now set M=M(g). Since g=ab then Hm(2, j) = M(a), for all j∈
supp2(M) and supp2(M) = supp(M(b)); so that ε2(M) = ∆(M(a)) and ω2(M) =
∆(M(b)). Analogously, ε1(M) = ∆(M(a)) and ω1(M) = ∆(M(b)). From this,
we get all assertions.
To sum up, from Proposition 31 we have obtained what kind of polynomials
we have to use to reach condition (3). This will be the main idea in order to
construct abelian codes C, with d(C) = ∆(C). We address this problem in the
following section.
7.1 Application 1: construction of abelian codes for which
its multivariate BCH bound, apparent distance and
minimum distance coincide.
In this section, we continue considering B={δBC H }and denoting ∆ = ∆δB CH ,
for the sake of simplicity by the same reasons given in the paragraphs prior
Proposition 29. Bearing in mind Proposition 34 and Proposition 27, we intro-
duce the following definition.
Definition 35 A matrix Pof order r1×r2, with entries in Lis called a com-
posed polynomial matrix (CP-matrix, for short) if there exist polynomials
a=a(X1)∈L(r1)and b=b(X2)∈L(r2)such that P=M(ab), where
ab ∈L(r1, r2).
22
Note that, for a CP-matrix P, its support is a direct product supp(P) =
supp(a(X1))×supp(b(X2)). The polynomials aand bare called the polynomial
factors of P. The reader may see that to check if a matrix is a CP-matrix is a
trivial task, because it must happen π1(supp(P)) = supp(a) = supp(M(a)) and
π2(supp(P)) = supp(b) = supp(M(b)). The following result is an immediate
consequence of Proposition 34.
Corollary 36 Let P=M(g)be a CP-matrix of order r1×r2with polynomial
factors aand b; that is, g=ab. If Xh1
1a|Xr1
1−1and Xh2
2b|Xr2
2−1, for some
h1, h2∈N, then
1. Z(ab) = Z(a)×Z(b).
2. ∆(P) = ∆(M(a)) ·∆(M(b)) = Z(g).
3. ∆1(P) = ∆2(P) = ∆(P) = Z(g)(the condition (3)).
4. ∆(M(a)) = ε2(P) = ω1(P)and
5. ∆(M(b)) = ε1(P) = ω2(P).
Example 37 Set q= 2, r1= 3 and r2= 7; so that n= 21. Let Pbe the CP-
matrix with polynomial factors a=X1+X2
1and b=X2+X2
2+X4
2, and g=ab.
In this case, X2
1a|X3
1−1 and X6
2b|X7
2−1. Now ∆(M(a)) = ∆(M(X2
1a)) = 2,
∆(M(b)) = ∆(M(X6
2b)) = 4; hence ∆(P) = 8 = |Z(g)|.
The next example shows that the hypothesis on the polynomials aand bof
Corollary 36 are not superfluous.
Example 38 Set q= 2, r1= 5 and r2= 7; so that n= 35. Let Pbe the
CP-matrix with factors a=X1+X2
1+X3
1+X4
1and b=X2+X2
2+X4
2. In
this case, Xh1
1a∤X3
1−1, for all h1∈Z5. On the other hand X6
2b|X7
2−1. Now
∆(M(a)) = 2, ∆(M(b)) = 4. Although ∆1(P) = ∆2(P) = ∆(P) = 8, one may
check that Z(ab)= 16.
Now we give a method for constructing the desired abelian codes. First, a
technical lemma.
Lemma 39 Let D⊂Zr1×Zr2be union of q-orbits and M=M(D), the matrix
afforded by D. If supp(M) = π1(supp(M)) ×π2(supp(M)) then B−mad =
∆B(M), where Bis any set of ds-bounds.
In the case B={δBCH }, the above equality coincides with the multivariate
BCH bound in [3, Theorem 30].
Proof. Clearly, in this case all rows (columns) have the same support and so
if one row or column is involved then all of them are too. The last assertion
comes directly from the computation of the multivariate BCH bound.
23
Remark 40 We have already mentioned that for any CP-matrix, M, one has
supp(M) = π1(supp(M)) ×π2(supp(M)). The converse is true for those ma-
trices satisfying hypothesis of lemma above; that is, if D⊂Zr1×Zr2is union
of q-orbits and M=M(D) is the matrix afforded by Dwith supp(M) =
π1(supp(M)) ×π2(supp(M)) then Mis a CP-matrix.
Theorem 41 Let Kbe an intermediate field Fq⊆K⊆L,a=a(X1)∈K(r1)
and b=b(X2)∈K(r2)be such that a|Xr1
1−1and b|Xr2
2−1. If there
exist (α1, α2)∈U,h1∈Zr1and h2∈Zr2for which ϕ−1
α1,Xh1
1a∈Fq(r1)
and ϕ−1
α2,Xh2
2b∈Fq(r2), then the abelian code C=ϕ−1
α1,Xh1
1a·ϕ−1
α2,Xh2
2bin
Fq(r1, r2)verifies ∆ (M(ab)) = ∆(C) = d(C).
Moreover, in this case, for any β1∈Ur1and β2∈Ur2the abelian code
C(β1,β2)=ϕ−1
β1,Xh1
1a·ϕ−1
β2,Xh2
2bis an ideal of Fq(r1, r2)and verifies ∆ (M(ab)) =
∆(C(β1,β2)) = d(C(β1,β2)) = d(C).
Proof. Set g(X1, X2) = Xh1
1a(X1)·Xh2
2b(X2) and α= (α1, α2). By def-
inition of the discrete Fourier transform, it is easy to see that the particular
factorization of gimplies that ϕ−1
(α1,α2),g =ϕ−1
α1,Xh1
1a·ϕ−1
α2,Xh2
2b. On the other
hand, it is clear that M(g) is a CP-matrix satisfying the hypothesis of Corol-
lary 36. This, in turn, implies that statement 2(b) of Theorem 25 is satisfied.
Let Mbe the matrix afforded by Dα(C). Since C=ϕ−1
α,g then supp(M) =
supp(M(g)), hence Mis also a CP-matrix and ∆(M) = ∆(M(g)). By Lemma 39,
B−mad(M) = ∆(M) and so statement 2(a) of Theorem 25 is also satisfied.
Thus ∆(C) = d(C). The final assertion is a direct consequence of [9, Remark
2] together with the fact that, under these hypothesis, all afforded matrices are
CP-matrices.
Now, we may apply all known criteria for univariate polynomials to have in-
verse of the discrete Fourier transform in an specific quotient ring. The following
corollary concretizes the proposed construction. It comes from [9, Remark 2]
and the theorem above.
Corollary 42 Let Kbe an intermediate field Fq⊆K⊆L,a=a(X1)∈K(r1)
and b=b(X2)∈K(r2)be such that a|Xr1
1−1and b|Xr2
2−1. If there exist
(α1, α2)∈U,h1∈Zr1and h2∈Zr2for which hXh1
1a(αi
1)iq
=Xh1
1a(αi
1),
for all i∈ {0, . . . , r1−1}and hXh2
2b(αj
2)iq
=Xh2
2b(αj
2), for all j∈
{0,...,r2−1}, then the family of abelian codes
C(β1,β2)=ϕ−1
β1,Xh1
1a·ϕ−1
β2,Xh2
2b|β1∈Ur1and β2∈Ur2
in Fq(r1, r2)verifies ∆ (M(ab)) = ∆(C(β1,β2)) = d(C(β1,β2)).
24
The following example shows how to use Corollary 42.
Example 43 Set q= 2, r1= 3 and r2= 45 (so n= 135). Fix α1∈U3and
α2∈U45. Consider the polynomials a=X+ 1 and b=Y40 +Y39 +Y38 +Y36 +
Y35 +Y32 +Y30 +Y25 +Y24 +Y23 +Y21 +Y20 +Y17 +Y15 +Y10 +Y9+Y8+
Y6+Y5+Y2+ 1. Then a|X3−1. Note that supp (X·a(X)) = {1,2}=C2(1)
modulo 3. By [9, Lemma 1] we have that h1= 1 works. Now, the polynomial b
appears in [9, Example 5] where it was mentioned that b|x45 −1 in F2[x] (so
that K=F2). In that example, it is shown that, for α2∈U45 (for instance, the
one with minimal polynomial Y12 +Y3+1), since b(1) = 1 and b(α3
2) = α30
2, then
(Y5b)(1) = 1, (Y5b)α3
2= (α3
2)5α30
2=α45
2= 1. So h2= 5 will work because
(Y5b)α6
2= (Y5b)α12
2= (Y5b)α24
2= 1; note that C2(3) = {3,6,12,24}
modulo 45. Now set C= (ϕ−1
α1,Xa ·ϕ−1
α2,Y 5b)⊂F2(r1, r2) = F2(3,45). Then
D(α1,α2)(C) = C2(1) ×(C2(1) ∪C2(3) ∪C2(9) ∪C2(21)).
One may check that 10 = ∆(M(ab)); so that d(C) = 10 and dimF2(C) = 87.
The next example shows that from a code satisfying the conditions of The-
orem 25, we can obtain a code with better parameters by making slight modi-
fications on the defining set in such a way that the new code verifies the same
conditions, but it has higher dimension, for example.
Example 44 Set q= 2, r1= 3 and r2= 45. Fix α1∈U3and α2∈
U45. Consider the code Cin Example 43; that is D(α1,α2)(C) = C2(1) ×
(C2(1) ∪C2(3) ∪C2(9) ∪C2(21)) and set g=Xa ·Y5b. As one may check
there are three subsets determining ∆(M(g)); to witt
S1={(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)},
S2={(1,16),(1,17),(1,18),(1,19),(2,16),(2,17),(2,18),(2,19)}and
S3={(1,31),(1,32),(1,33),(1,34),(2,31),(2,32),(2,33),(2,34)}.
If one computes ∆(M(g)) by considering S1then clearly C2(1) ×C2(21) will
have no influence in the computation. Hence one may construct the new code
C′for which D(α1,α2)(C′) = C2(1)×(C2(1) ∪C2(3) ∪C2(9)) such that ∆(C′) =
∆(C) = ∆(M(g)).
Note that the matrix afforded by D=D(α1,α2)(C′) is also a CP-matrix
and so B−mad(M(D)) = ∆(M(D)) = 10. Since Cis a subcode of C′then c=
ϕ−1
α1,g ∈C′and, clearly, gsatisfies conditions (2a) and (2b) of Theorem 25 for C′;
hence ∆(C′) = ∆(M(g)) = d(C′) = 10 = d(C) = ∆(C). Since dimF2(C′) = 95
and dimF2(C) = 87, C′is a code with better parameters than those of C.
Next application comes from [9, Corollary 6].
Corollary 45 Let Kbe an intermediate field Fq⊆K⊆Land a=a(X1)∈
K(r1)be such that a|Xr1
1−1with ϕ−1
α1,Xh1
1a∈Fq(r1), for some α1∈Ur1and
h1∈Zr1.
Let gbe an irreducible factor of Xr2
2−1in K[X2]with defining set Dα2(g),
for some α2∈Ur2. Set b= (Xn
2−1)/g. If there are positive integers j, t such
25
that b(αj
2) = αt
2and gcd j, r2
gcd(q−1,r2)|t, then there exists h2∈Zr2such that
the abelian code C=ϕ−1
α1,Xh1
1a·ϕ−1
α2,Xh2
2bin Fq(r1, r2)verifies ∆ (M(ab)) =
∆(C) = d(C).
Our last application of this section is the following result that comes from
[9, Corollary 7].
Corollary 46 Let a=a(X1)∈L(r1)be such that a|Xr1
1−1with ϕ−1
α1,Xh1
1a∈
F2(r1), for some α1∈Ur1and h1∈Zr1, and suppose r2= 2m−1, for some
m∈N. Then there exist at least φ(r2)
mbinary codes Cof length n=r1r2such
that ∆ (M(ab)) = ∆(C) = d(C).
In the following examples we take the advantage of the information for cyclic
codes from the tables appearing in [9]
Example 47 We first show in Table 1 some abelian codes of lenght 7×15 = 105
constructed from a list of divisors aiof X7
1−1 in F2[X1] and divisors bjof X15
2−1
in F2[X2] as in Corollary 45. The divisors are a1= 1 + X1,a2= 1 + X1+X3
1,
a3= 1 + X2
1+X3
1,b1=X15
2−1
1+X2+X2
2
,b2=X15
2−1
1+X2+X4
2
and b3=X15
2−1
1+X3
2+X4
2
.
a h1b h2Dimension ∆ = d
a21b11 30 8
a21b21 24 16
a21b33 24 16
a33b11 30 8
a33b21 24 16
a33b33 24 16
a1a30b11 40 6
a1a30b21 32 12
a1a30b33 32 12
a2a30b11 70 2
a2a30b21 56 4
a2a30b33 56 4
Table 1: Abelian Codes in F2(7,15).
In Table 2 we also have abelian codes of lenght 105, but in this case we
consider r1= 5 and r2= 21. Here we choose only one divisor of X5
1−1 in
F2[X1]; the 5-th cyclotomic polynomial, Φ5. The other divisors b′
jof X21
2−1
in F2[X2] from which we may construct abelian codes as in Corollary 45 are
b′
1=X21
2−1
1+X2+X2
2
,b′
2=X21
2−1
1+X2+X3
2
,b′
3=X21
2−1
1+X2
2+X3
2
,b′
4=X21
2−1
1+X2+X2
2+X4
2+X6
2
and
b′
5=X21
2−1
1+X2
2+X4
2+X5
2+X6
2
.
26
a h1b h2Dimension ∆ = d
Φ50b′
11 70 2
Φ50b′
21 60 3
Φ50b′
33 60 3
Φ50b′
41 40 6
Φ50b′
51 40 6
Table 2: Abelian Codes in F2(5,21).
7.2 Application 2: True distance in BCH multivariate
codes
In [3, Definition 33], the notion of BCH multivariate code appears. Let us recall
this definition focused on the bivariate case.
Definition 48 Let ¯γ⊆ {1,2}and ¯
δ={(δk)k∈¯γ|2≤δk≤rk}. An abelian
code Cin Fq(r1, r2)is a bivariate BCH code of designed distance ¯
δif
there exists a list of positive integers ¯
b= (bk)k∈¯γsuch that
Dα(C) = [
k∈¯γ
δk−2
[
l=0 [
i∈I(k,bk+l)
Q(i)
for some α∈U, where {bk, . . . , bk+δk−2}is a list of consecutive integers
modulo rkand I(k, u) = {i∈I|i(k) = u}.
The BCH multivariate codes are denoted Bq(α, ¯γ , ¯
δ, ¯
b).
Let Cbe an abelian code in Fq(r1, r2) with M=M(Dα(C)) the matrix
afforded by its defining set with respect to some ¯α= (α1, α2)∈U. If Msat-
isfies supp (M) = π1(supp(M)) ×π2(supp(M)) then Dα(C) = π1(supp(M)) ×
π2(supp(M)). We set S1=π1(supp(M)) and S2=π2(supp(M)). Then, one
may consider the cyclic codes C1and C2with defining sets D1=Zr1\S1and
D2=Zr2\S2with respect to α1and α2, respectively (note that it may happen
Dα(C)6=D1×D2).
Now suppose that the code Cis an abelian code as described in Theorem 41
keeping the notation for the polynomials aand band having in mind Remark 40.
By the proof of this theorem one also may deduce that viewing ϕ−1
α1,Xh1
1ain
Fq(r1) and ϕ−1
α2,Xh2
2bin Fq(r2) it happens that C1=ϕ−1
α1,Xh1
1a⊆Fq(r1)
and C2=ϕ−1
α2,Xh2
2b⊆Fq(r2). It is also clear that the cyclic codes C1and
C2verify that their minimum distances equal their respective maximum BCH
bounds, as aand bsatisfy the conditions in [9, Corollary 5].
In some sense we may consider C1and C2as “projected codes”, but clearly
Cis not a product of them under any classical algebraic operation; however, for
27
their defining sets the equality under the product occurs, and all properties of
ds-bounds are related, as the following results show.
Lemma 49 Under the same notation from previous paragraphs, let Cbe an
abelian code in Fq(r1, r2), with M=M(Dα(C)) and suppose supp (M) =
π1(supp(M)) ×π2(supp(M)). Consider D1=Zr1\π1(supp(M)),D2=
Zr2\π2(supp(M)) and let Cibe the cyclic code with Dαi(Ci) = Di, for i∈ {1,2}.
Then
1. For any set Bof ds-bounds, ∆B,α(C) = ∆B,α1(C1)·∆B,α2(C2).
2. Cis a nonzero BCH multivariate code if and only if C1and C2are BCH
cyclic codes in the classical sense (see [17]).
Moreover, if case (2) holds, with Ci= (αi, δi, bi), for i∈ {1,2}, then C=
Bq((α1, α2),{1,2},{δ1, δ2},{b1, b2}).
Proof. Assertion (1) comes from Corollary 36, having in mind Remark 40.
Now we prove assertion (2). First, suppose that Cis a multivariate BCH
code. Assume that 1 ∈¯γ, and let B={b1, . . . , b1+δ1−2}be its list of
consecutive integers modulo r1. Consider a q-cyclotomic coset T⊆D1and take
t∈T. Since t∈D1then the set (t, Zr2)⊆ Dα(C) (with the obvious meaning).
If (t, Zr2)∩Q(b1+l, Zr2)6=∅, for some l∈ {0,...,δ1−2}, then we are done.
Otherwise, it must happen 2 ∈¯γand (t, Zr2)⊂Sδ2−2
l=0 Q(Zr1,b2+l). Since
C6= 0 then we may take an element u∈Zr2\Sδ2−2
l=0 Cq(b2+l) and clearly
(t, u)6∈ Sδ2−2
l=0 Q(Zr1,b2+l), a contradiction.
The final assertion comes immediately from the fact that Mis a CP-matrix.
Theorem 50 Let Kbe an intermediate field Fq⊆K⊆L,a=a(X1)∈K(r1)
and b=b(X2)∈K(r2)be such that a|Xr1
1−1and b|Xr2
2−1. If there
exist (α1, α2)∈U,h1∈Zr1and h2∈Zr2for which ϕ−1
α1,Xh1
1a∈Fq(r1)and
ϕ−1
α2,Xh2
2b∈Fq(r2), with at least one of the inverses different from 1, then there
exists in Fq(r1, r2)a family of permutation equivalent BCH multivariate codes
nCβ=Bq(β, ¯γ , ¯
δ, ¯
b)|β∈Uosuch that
1. ¯γ⊆ {1,2}and
(a) If supp(a) = Zr1then 16∈ ¯γ.
(b) If supp(b) = Zr2then 26∈ ¯γ.
2. ¯
δ={δk|k∈¯γ}with δ1= ∆(M(a)) and δ2= ∆(M(b)).
3. Y
k∈¯γ
δk= ∆(Cβ) = d(Cβ), for each ¯
β∈U.
28
4. ϕ−1
β1,Xh1
1a·ϕ−1
β2,Xh2
2b=ϕ−1
β,X h1
1aXh2
2b∈Cβ, where β= (β1, β2).
Proof. Set ¯
b={bk|k∈¯γ}. First, suppose supp(M(a)) = supp(a)6=Zr1and
supp(M(b)) = supp(b)6=Zr2. Then by [9, Theorem 2], for each β∈U, there
exist BCH cyclic codes B(a) = Bq(β1, δ1, b1) and B(b) = Bq(β2, δ2, b2) such
that δ1= ∆(M(a)), δ2= ∆(M(b)), ϕ−1
β1,Xh1
1a∈B(a) and ϕ−1
β2,Xh2
2b∈B(b). Now
let C=Cβbe the abelian code with Dβ(C) = Dβ1(B(a)) ×Dβ2(B(b)) and set
M=MDβ(C). It is clear that Mis a CP-matrix and, moreover, following
the notation of Lemma 49, D1=Dβ1(B(a)) and D2=Dβ2(B(b)). Then Cis
a bivariate BCH code with ¯γ={1,2},¯
δ={δ1, δ2}and b={b1, b2}. Now by
statement (1) of Lemma 49, we have ∆β(C) = δ1δ2and clearly statement (5)
of this theorem holds.
It remains to prove the equality ∆(Cβ) = d(Cβ). On the one hand, we
have ∆β(C) = ∆(M(a))∆(M(b)) = ∆(M(ab)) and, on the other hand, by hy-
pothesis, ∆(M(ab)) = Z(ab). Hence, by applying Theorem 25, we are done.
We may take, again, advantage from cyclic codes to transform a given abelian
code C= (g), with d(C) = ∆(C) into another abelian code with higher dimen-
sion, as in Example 44, until to get a new BCH code.
Example 51 We continue with the code Cfrom Example 43. Recall that q= 2,
r1= 3, r2= 45 and we have fixed α1∈U3and α2∈U45. We have polynomials
a=X+ 1 and b=Y40 +Y39 +Y38 +Y36 +Y35 +Y32 +Y30 +Y25 +Y24 +
Y23 +Y21 +Y20 +Y17 +Y15 +Y10 +Y9+Y8+Y6+Y5+Y2+ 1 such that
a|X3
1−1 and h1= 1 works, and b|X45
2−1 and h2= 5 works in the sense of
Theorem 50. Hence the hypothesis of this theorem are satisfied.
Now we have to follow the proof to construct our multivariate BCH code.
The proof of Theorem 50 uses a construction from [9, Theorem 2]. Clearly
B(a) is the BCH code in F2(3) with defining set Dα1(B(a)) = C2(1). On the
other hand, by [9, Example 8], the code B(b) has defining set Dα2(B(b)) =
C2(1) ∪C2(3) so that it is a BCH code. In fact, B(b) = B2(α2,5,1), following
the usual notation for BCH codes.
Thus C(α1,α2)=B2((α1, α2),{1,2},{2,5},{1,1}), dC(α1,α2)= 10 and
dimF2C(α1,α2)= 58. This code has better parameters than the code Cfrom
Example 43 and C′from Example 44.
We finish by extending Corollary 45 to bivariate BCH codes.
Corollary 52 Let Kbe an intermediate field Fq⊆K⊆Land a=a(X1)∈
L(r1)be such that a|Xr1
1−1, with ϕ−1
α1,Xh1
1a∈Fq(r1), for some α1∈Ur1and
h1∈Zr1.
Let gbe an irreducible factor of Xr2
2−1in K[X2]with defining set Dα2(g),
for some α2∈Ur2. Set b= (Xn
2−1)/h. If there are positive integers j, t such
that b(αj
2) = αt
2and gcd j, r2
gcd(q−1,r2)|tthen there exists a bivariate BCH
29
code C=Bq(α, ¯γ, ¯
δ, ¯
b)in Fq(r1, r2)verifying ∆ (M(ab)) = ∆(C) = d(C), for
certain α, ¯γ , ¯
δ, ¯
b.
Proof. Comes immediately from Corollary 45 together with Theorem 50.
Example 53 We shall extend to bivariate BCH codes those abelian codes on
Table 1 from Example 47. Recall that we had a list of divisors aiof X7
1−1
in F2[X1] and divisors bjof X15
2−1 in F2[X2], namely, a1= 1 + X1,a2=
1+X1+X3
1,a2= 1+X2
1+X3
1,b1=X15
2−1
1+X2+X2
2
,b2=X15
2−1
1+X2+X4
2
and b3=X15
2−1
1+X3
2+X4
2
from which we constructed the mentioned table.
Now, one may check easily that the codes determined by a2,a3and a1a3
are all BCH. Specifically, B(a2) = B2(α1,4,5), B(a3) = B2(α1,4,0) and
B(a1a3) = B2(α1,3,5), while the code determined by a2a3is all F2(7). On
the other hand, as it is shown in [9, Example 9] one may construct from b1the
code B(b1) = B2(α1,2,0), of dimension 14, from b2the code B2(α1,4,13) of
dimension 10, and from b3the code B2(α1,4,0) which also has dimension 10.
Thus, Table 3 is the new table of bivariate BCH codes in F2(7,15):
¯γ¯
bDimension ∆ = d
{1,2} {5,0}42 8
{1,2} {5,13}40 16
{1,2} {5,0}40 16
{1,2} {0,0}42 8
{1,2} {0,13}40 16
{1,2} {0,0}40 16
{1,2} {5,0}56 6
{1,2} {5,13}40 12
{1,2} {5,0}40 12
{2} {0}98 2
{2} {13}70 4
{2} {0}70 4
Table 3: Bivariate BCH codes in F2(7,15):
In the case of codes in F2(5,21), the code determined by a= Φ5is F2(5), so
we construct B(b′
1) = B2(α2,2,0), B(b′
2) = B2(α2,3,19), B(b′
3) = B2(α2,3,1),
B(b′
4) = B2(α2,6,17) and B(b′
5) = B2(α2,6,0). Table 4 is the new table of
bivariate BCH codes in F2(5,21).
30
¯γ¯
bDimension ∆ = d
{2} {0}100 2
{2} {19}75 3
{2} {1}75 3
{2} {17}55 6
{2} {0}55 6
Table 4: Bivariate BCH codes in F2(5,21).
8 Conclusion
We have developed a technique to extend any bound for the minimum distance
of cyclic codes constructed from its defining sets (ds-bounds) to abelian (or
multivariate) codes through the notion of B-apparent distance. We used this
technique to improve the searching for new bounds for abelian codes having
unknown minimum distance. We have also studied conditions for an abelian
code to verify that its B-apparent distance reaches its (true) minimum distance
and we have constructed some tables of such codes as an application.
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