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arXiv:math/0310148v1 [math.OC] 10 Oct 2003
CONVOLUTIONAL CODES OF GOPPA TYPE
J.A DOM´
INGUEZ P´
EREZ, J.M MU ˜
NOZ PORRAS, AND G. SERRANO SOTELO
Abstract. A new kind of Convolutional Codes generalizing Goppa Codes is pro-
posed. This provides a systematic method for constructing convolutional codes with
prefixed properties. In particular, examples of Maximum-Distance Separable (MDS)
convolutional codes are obtained.
1. Introduction
The aim of this paper is to propose a definition of Convolutional Goppa Codes
(CGC). This definition will provide an algebraic method for constructing Convolutional
Codes with prescribed invariants.
We propose a definition of CGC in terms of families of curves X→A1parametrized
by the affine line A1= Spec Fq[z] over a finite field Fq. In this setting, the usual
definition of a Goppa Code as the code obtained by evaluation of sections at several
rational points, is translated as a code obtained by evaluation (of sections of some
invertible sheaf over X) along several sections of the fibration X→A1.
The paper is organized as follows.
In §2 we offer a summary on Goppa Codes following [3], [6], and using the standard
notations of Algebraic Geometry [2].
§3 is devoted to giving the general definition of CGC and gives some general results.
In §4 we study the case of a trivial fibration of projective lines over A1and we
conclude giving some explicit examples of MDS convolutional codes.
We freely use the standard notations of abstract Algebraic Geometry as can be found
in [2]. After the works of V. Lomadze [4], J. Rosenthal and R. Smarandache [7], [8],
there is evidence that the use of methods of Algebraic Geometry can be relevant to the
study of Convolutional Codes. This paper is a step in favor of that evidence.
2. Background on Algebraic Geometry and Goppa Codes
In this Section we summarize the basic definitions about Goppa Codes, constructed
using methods of Algebraic Geometry (see [3], [6]).
Let Xbe a geometrically irreducible, smooth and projective curve over the finite
field Fq. Let p1,...,pnbe ndifferent Fq-rational points of X, and Dthe divisor
D=p1+···+pn. Let Gbe another effective divisor with support disjoint from D.
The Goppa code C(G, D) defined by (G, D) is the linear code of length nover Fq
defined as the image of the linear map
α:L(G)→Fn
q
f7→ (f(p1),...,f(pn)) ,
Key words and phrases. Convolutional Codes, Goppa Codes, MDS Codes, Algebraic Curves, Co-
herent Sheaves, Finite Fields.
This research was partially supported by the Spanish DGESYC through research project BMF2000-
1327 and by the “Junta de Castilla y Le´on” through research projects SA009/01 and SA032/02.
1
2 J.A DOM´
INGUEZ P´
EREZ, J.M MU ˜
NOZ PORRAS, AND G. SERRANO SOTELO
where L(G) is the complete linear series defined by G. That is, let Fq(X) be the field
of rational functions over the curve X,
L(G) = {f∈Fq(X) such that Div(f) + G≥0}.
The Goppa code has dimension
k= dim C(G, D) = dim L(G)−dim L(G−D).
Let gbe the genus of X; if we assume the inequality 2g−2<deg(G)< n, then one
has
k= deg(G)−g+ 1 ,
and the minimum distance dof C(G, D) satisfies the inequality
d≥n−deg(G).
Let OX(D) be the invertible sheaf on Xdefined by the divisor D. One has the following
exact sequence of sheaves
0→ OX(−D)→ OX→ OD→0,
where OD≃ Op1/mp1× · · · × Opn/mpn≃Fq×n)
...×Fq. Tensoring the above exact
sequence by OX(G), one obtains
0→ OX(G−D)→ OX(G)→ OD→0.
By taking global sections, we obtain an exact sequence of cohomology
0→H0(X, OX(G−D)) →H0(X, OX(G)) α
→ OD→H1(X, OX(G−D)) →
→H1(X, OX(G)) →0,
where L(G) = H0(X, OX(G)) and αis the evaluation map defined above.
In the case 2g−2<deg(G)< n, one has the exact sequence
(2.1) 0 →H0(X, OX(G)) α
→ OD→H1(X, OX(G−D)) →0.
Let ωXbe the dualizing sheaf of X, which is isomorphic to the sheaf of regular
1-forms over X;H0(X, ωX) is the Fq-vector space of global regular 1-forms over X,
which is of dimension g= genus of X.
By Serre’s duality ([2]), there exist canonical isomorphisms of Fq-vector spaces
H1(X, L)∗≃H0(X, ωX⊗ L−1)
for every invertible sheaf Lon X. Given a divisor Dover X, we shall denote by Ω(D)
the vector space H0(X, ωX⊗ OX(−D)).
The dual Goppa code, C∗(G, D), associated with the Goppa code C(G, D) is defined
as the linear code of length nover Fqgiven by the image of the linear map
α∗: Ω(G−D)→Fn
q
η7→ (Resp1(η),...,Respn(η)) ,
Let us take duals in the exact sequence (2.1):
0→H1(X, OX(G−D))∗β
→ O∗
D
αt
→H0(X, OX(G))∗→0.
By Serre’s duality, one has isomorphisms
H1(X, OX(G−D))∗≃Ω(G−D),
H0(X, OX(G))∗≃H1(X, ωX⊗ OX(−G)) ,
CONVOLUTIONAL CODES OF GOPPA TYPE 3
and the above sequence is the cohomology sequence induced by the exact sequence of
sheaves
0→ωX(−G)→ωX(D−G)→ωX(D−G)⊗OXOD→0,
where we denote ωX(−G) = ωX⊗ OX(−G), and βis precisely the map α∗defining
C∗(G, D).
Given a linear series Γ ⊆H0(X, OX(G)), that is, a vector subspace defining a family
of divisors linearly equivalent to G, we define the Goppa code C(Γ, D) associated whit
Γ and Das the image of the homomorphism α|Γ:
H0(X, OX(G)) α//OD
S|
Γ
α|Γ
77
p
p
p
p
p
p
p
p
p
p
p
p
p
p
When Γ &H0(X, OX(G)), we shall say that C(Γ, D) is a non-complete Goppa code.
3. Convolutional Goppa Codes
We shall contruct a kind of convolutional code that generalizes the notion of Goppa
codes. These codes will be associated with families of algebraic curves.
Given an algebraic variety Sover Fq, a family of projective algebraic curves parametrized
by Sis a morphism of algebraic varieties π:X→S, such that πis a projective and flat
morphism whose fibres Xs=π−1(s) are smooth and geometrically irreducible curves
over Fq(s) (the residue field of s∈S).
Let us consider a family of curves Xπ
→Uparametrized by U= Spec Fq[z] = A1.
Given a closed point u∈Uwith residue field Fq(u), the fibre Xu=π−1(u) is a curve
over the finite field Fq(u).
Let pi, 1 ≤i≤n, be ndifferent sections, pi:U→X, of the projection π. These
sections define a Cartier divisor on X:
D=p1(U) + ···+pn(U),
which is flat of degree nover the base U([2]).
Note that given a coherent sheaf Fon X, the cohomology groups Hi(X, F) are finite
Fq[z]-modules and Hi(X, F) = 0 for i≥0 (see [2] III).
Let Lbe an invertible sheaf over X. One has an exact sequence of sheaves on X
(3.1) 0 → L(−D)→ L → OD→0,
which induces a long exact cohomology sequence
(3.2)
0→H0(X, L(−D)) →H0(X, L)α
→H0(X, OD)→H1(X, L(−D)) →H1(X, L)→0.
Let rbe the degree of Lin each fibre of π(which is independent of the fibre) and
let gbe the genus of any fibre of π(also independent of the fibres).
Proposition 3.1. Let us assume that 2g−2< r. Then, one has that H1(X, L) = 0
and H0(X, L)is a free Fq[z]-module of rank r−g+ 1
Proof. Under the condition 2g−2< r, one has that H1(Xu,L|Xu) = 0 for every point
u∈U. Note that Hi(X, F)e=Riπ∗Ffor every coherent sheaf Fon X([2] III), and
applying ([2] III Corollary 12.9) one concludes the proof.
4 J.A DOM´
INGUEZ P´
EREZ, J.M MU ˜
NOZ PORRAS, AND G. SERRANO SOTELO
Under the hypothesis of Proposition 3.1, there exists an exact sequence of Fq[z]-
modules
(3.3) 0 →H0(X, L(−D)) →H0(X, L)α
→H0(X, OD)→H1(X, L(−D)) →0.
where H0(X, OD) is a free Fq[z]-module of rank n.
Remark 3.2.Let η∈Ube the generic point of U, whose residue field is Fq(z); the fibre
Xη=π−1(η) is a smooth, irreducible curve over Fq(z). Note that p1(η),...,pn(η) are n
different Fq(z)-rational points of the curve Xη. One then has a canonical decomposition
of H0(X, OD)ηas a Fq(z)-algebra
H0(X, OD)η=Fq(z)×n)
...×Fq(z).
Given a Fq[z]-module M, let us denote by Mηthe Fq(z)-vector space
Mη=M⊗Fq[z]Fq(z).
The sequence (3.3) induces an exact sequence of Fq(z)-vector spaces
(3.4) 0 →H0(X, L(−D))η→H0(X, L)η
αη
→H0(X, OD)η→H1(X, L(−D))η→0.
Definition 3.3. The complete convolutional Goppa code associated with Land Dis
the image of the homomorphism αη
C(L, D) = ImH0(X, L)η
αη
−→ H0(X, OD)η≃Fq(z)n.
Given a free submodule Γ ⊆H0(X, L), the convolutional Goppa code associated with
Γ and Dis the image of αη|Γη
C(Γ, D) = ImΓη
αη
−→ Fq(z)n.
Remark 3.4.We use definition 2.4 of [5] as definition of convolutional codes. Any
matrix defining αη(respectively αη|Γη) is a generator matrix of rational functions for
the code C(L, D) (resp. C(Γ, D)).
The canonical decomposition H0(X, OD)η≃Fq(z)nas Fq(z)-algebras does not ex-
tend (in general) to a decomposition H0(X, OD)≃Fq[z]nas rings. In fact, one has a
canonical isomorphism of rings H0(X, OD)φ
∼
→Fq[z]nonly when p1(U),...,pn(U) are
disjoint sections. However, H0(X, OD) is a free Fq[z]-module; then, there exist (non-
canonical) isomorphisms of Fq[z]-modules:
H0(X, OD)φ
∼
→Fq[z]⊕n)
...⊕Fq[z],
which are not (in general) isomorphism of rings.
This allows us to give another definition of convolutional Goppa codes.
Definition 3.5. Given a trivialization φ:H0(X, OD)∼
→Fq[z]nas Fq[z]-modules, one
defines the convolutional Goppa code C(L, D, φ) as the image of φ◦α
H0(X, L)α
→H0(X, OD)φ
∼
→Fq[z]n.
Anagously, one defines the convolutional Goppa code C(Γ, D, φ).
CONVOLUTIONAL CODES OF GOPPA TYPE 5
Let us assume (for the rest of the paper) that the invariants (r, n, g) satisfy the
inequality
2g−2< r < n .
Proposition 3.6. Under the above conditions on (r, n, g),H0(X, L(−D)) = 0 and
H1(X, L(−D)) is a free Fq[z]-module. The following exact sequence is exact
(3.5) 0 →H0(X, L)α
→H0(X, OD)→H1(X, L(−D)) →0.
and remains exact when we take fibres over every point u∈U.
Proof. If 2g−2< r < n,H0(Xu,L(−D)|Xu) = 0 for every point u∈U; and applying
([2] III Corollary 12.9) one concludes.
Corollary 3.7. The convolutional code C(L, D, φ)has dimension k=r−g+ 1 and
length n. Every matrix defining φ◦αis a basic generator matrix [5] for C(L, D, φ).
Proof. This is a direct consecuence of the last statement of Proposition 3.6 and the
characterization of basic generator matrices of [5].
Let us consider the convolutional Goppa code C(Γ, D, φ) defined by a submodule
Γ⊆H0(X, L) and a trivilization φ. With the above restrictions, one has:
Proposition 3.8. Every matrix defining φ◦α|Γis a basic generator matrix for the
code C(Γ, D, φ)if and only if H0(X, L)/Γis a torsion-free Fq[z]-module.
Proof. The sequence (3.5) induces a diagram
0
0
0//Γ
α|Γ//H0(X, OD)//H1(X, Γ)
//0
0//H0(X, L)
//H0(X, OD)//H1(X, L(−D))
//0
H0(X, L)/Γ0
Then, the kernel of H1(X, Γ) →H1(X, L(−D)) is isomorphic to H0(X, L)/Γ and
H1(X, L(−D)) is free. This implies that the torsion elements of H1(X, Γ) are contained
in H0(X, L)/Γ, from which one concludes the proof.
The above results allow us to construct basic generator matrices for the codes
C(Γ, D, φ). If p1(U),...,pn(U) are disjoint sections and φthe canonical trivializa-
tion, this gives us a basic generator matrix for C(Γ, D). However, in general the codes
C(Γ, D) and C(Γ, D, φ) are different.
Let us describe a geometric way to obtain a basic generator matrix for C(L, D) and
C(Γ, D).
Assume that the curves p1(U),...,pn(U) meet transversally at some points, and let
¯
Xbe the blowing-up [2] of Xat these points. One has morphisms
¯
Xβ//
¯π=π◦β
@
@
@
@
@
@
@X
π
U
6 J.A DOM´
INGUEZ P´
EREZ, J.M MU ˜
NOZ PORRAS, AND G. SERRANO SOTELO
such that the proper transform of Dunder πis a divisor ¯
D⊂¯
Xsatisfying
¯
D=p1(U)∐ · · · ∐ pn(U)β
→D ,
and one has a canonical homomorphism of rings
0→ OD→β∗O¯
D
which induces
0→π∗OD
β
→¯π∗O¯
D≃
∼
Fq[z]n,
where ¯π∗O¯
D≃
∼
Fq[z]nis the canonical isomorphism of sheaves of rings.
β∗Lis an invertible sheaf on ¯
Xand there exists a canonical homomorphism
β∗L → O ¯
D→0,
whose kernel is (β∗L)(−¯
D). This induces
0→ L → β∗β∗L → β∗O¯
D,
and taking global sections one obtains
0→H0(X, L)γ
→H0(X, β∗β∗L)µ
→Fq[z]n.
The image of µis precisely a free submodule of Fq[z]nthat defines a basic generator
matrix for C(L, D).
Let us consider the sequence of homomorphisms
0→H0(X, L)α
→H0(X, OD)β
֒→H0(X, O¯
D) = Fq[z]n.
β◦αis not in general a basic matrix, since H0(X, O¯
D)/H0(X, OD) has torsion. Let
us define
¯
H0(X, L) = {p∈Fq[z]nsuch that λp ∈H0(X, L) for some λ∈Fq[z]}.
¯
H0(X, L)/H0(X, L) is a torsion module and Fq[z]n/¯
H0(X, L) is torsion-free. Then,
every matrix defining the homomorphism ¯
H0(X, L)֒→Fq[z]nis a basic generator
matrix for C(L, D).
This is an algebraic-geometric interpretation of Forney’s construction of the basic
matrices of a convolutional code [1].
4. Convolutional Goppa Codes associated with the projective line
Let P1= Proj Fq[x0, x1] be the projective line over Fq, and
X=P1×Uπ
→U= Spec Fq[z]
the trivial fibration. Let us denote by t=x1/x0the affine coordinate in P1, and by p∞
its infinity point. Let us consider the following ndifferent sections of π
pi:U→P1×U
defined in the coordinates (t, z) by
pi(z) = (αiz+βi, z), αi, βi∈Fq.
Let D=p1(U) + ···+pn(U) and let Lbe the invertible sheaf on X
L=π∗
1OP1(rp∞)⊗FqOU, r < n ,
CONVOLUTIONAL CODES OF GOPPA TYPE 7
The exact sequence (3.5) is in this case:
0→H0(X, L)α//H0(X, OD)//H1(X, L(−D)) //0.
|| ||
H0(P1,OP1(rp∞)) ⊗Fq[z]α//Fq[z]n
Taking the fibres over the generic point η, and the canonical trivialization (π∗OD)η≃
Fq(z)n, the homomorphism αηis the evaluation map at the points p1(η),...,pn(η)
αη:H0(P1,OP1(rp∞)) ⊗FqFq(z)→Fq(z)n
αη(tj) = tj(p1(η)),...,tj(pn(η))=(α1z+β1)j,...,(αnz+βn)j,
where {1,t,...,tr}is the “canonical” basis of H0(P1,OP1(rp∞)) in the affine coordinate
t. The convolutional code C(L, D) is a kind of generalized Reed-Solomon (RS) code (for
z= 0 we obtain a classical RS-code).
Let Γ ⊆H0(P1,OP1(rp∞)) be the linear subspace generated by {ts,...,tr}. The
convolutional Goppa code C(Γ, D) is the image of the homomorphism
αη: Γ⊗FqFq(z)→Fq(z)n
tj7−→ αη(tj),for s≤j≤r .
In this case H0(X, L)/Γ≃(H0(P1,OP1(rp∞))/Γ) ⊗FqFq[z] is torsion-free. Then, by
Proposition 3.8 every matrix defining
α: Γ ⊗FqFq[z]→H0(X, OD)
is a basic generator matrix. To compute a matrix for αexplicitly, we need to fix an
isomorphism of Fq[z]-modules
H0(X, OD)φ
→Fq[z]n,
and this gives a generator matrix for C(Γ, D, φ). However, it would be desirable to com-
pute basic matrices for the codes C(Γ, D). We shall do this in general in a forthcoming
paper. Here we shall offer some explicit examples.
Example 4.1.Let a, b ∈Fqbe two different non-zero elements, and
pi(z) = (ai−1z+bi−1, z), i = 1,...,n, with n < q .
The evaluation map αηover Γ is defined by the matrix
(4.1)
(z+ 1)s(az +b)s(a2z+b2)s... (an−1z+bn−1)s
(z+ 1)s+1 (az +b)s+1 (a2z+b2)s+1 ... (an−1z+bn−1)s+1
.
.
..
.
..
.
.....
.
.
(z+ 1)r(az +b)r(a2z+b2)r... (an−1z+bn−1)r
.
This matrix is a generator matrix for the code C(Γ, D). Using this construction we can
give concrete examples of CGC of dimension k=r−s+ 1 that are Maximum-Distance
Separable (MDS) convolutional codes, i.e., whose free distance attains the generalized
Singleton bound [7].
8 J.A DOM´
INGUEZ P´
EREZ, J.M MU ˜
NOZ PORRAS, AND G. SERRANO SOTELO
•If s=r, the convolutional Goppa code C(Γ, D) has dimension 1, degree r, and
(4.1) is a canonical (reduced and basic [5]) generator matrix. We can list a few
examples, where k/n,δand dare respectively the rate, the degree and the free
distance of the code.
field canonical generator matrix k/n δ d
F3={0,1,2}z+ 1 z+ 21/2 1 4
F4={0,1, α, α2}z+ 1 z+α z +α21/3 1 6
where α2+α+ 1 = 0
F5={0,1,2,3,4}(z+ 1)2(z+ 2)2(z+ 4)21/3 2 9
In these examples the sections p1,...,pnare disjoint, such that one has C(Γ, D) =
C(Γ, D, φ), where φ:H0(X, OD)∼
→Fq[z]nis the corresponding canonical trivi-
alization.
•If s < r, let us take a∈Fqas a primitive element.
Now, the matrix (4.1) is reduced, since the matrix of highest-degree terms in
each row is a Vandermonde matrix of rank k. The sections p1,...,pnare not
disjoint, but in some cases the matrix (4.1) is actually basic and we do not have
to find an isomorphism of Fq[z]-modules, φ:H0(X, OD)∼
→Fq[z]n, in order to
compute a basic generator matrix for the code C(Γ, D).
We present two examples of this situation.
field canonical generator matrix k/n δ d
F41 1 1
z+ 1 αz +α2α2z+α2/3 1 3
F5z+ 1 2z+ 3 4z+ 4 3z+ 2
(z+ 1)2(2z+ 3)2(4z+ 4)2(3z+ 2)21/2 3 8
Acknowledgments. We thank F.J. Plaza Mart´ın and E. G´omez Gonz´alez for many
enlightening comments, and J. Prada Blanco and J.I. Iglesias Curto for helpful ques-
tions that helped us to improve this paper.
CONVOLUTIONAL CODES OF GOPPA TYPE 9
References
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(1970) 720–738.
[2] R. Hartshorne, Algebraic Geometry Grad. Texts in Math., vol. 52, (Springer-Verlag, New York,
1977).
[3] T. Høholdt, J.H. van Lint and R. Pellikaan, Algebraic Geometric Codes, in: Handbook of Coding
theory, Ed. by V.S. Pless and W.C. Huffman (Elsevier, Amsterdam, 1998) 871–962.
[4] V. Lomadze, Convolutional Codes and Coherent Sheaves, AAECC 12 (2001) 273–326.
[5] R.J. McEliece, The Algebraic Theory of Convolutional Codes, in: Handbook of Coding theory,
Ed. by V.S. Pless and W.C. Huffman (Elsevier, Amsterdam, 1998) 1065–1138.
[6] J.H. van Lint and G. van der Geer, Introduction to Coding Theory and Algebraic Geometry DMV
Seminar, vol. 12, (Birkh¨auser, Basel, 1998).
[7] J. Rosenthal and R. Smarandache, Maximum Distance Separable Convolutional Codes, AAECC
10 (1999) 15–32.
[8] R. Smarandache and J. Rosenthal, Constructions of MDS-Convolutional Codes, IEEE Trans.
Inform. Theory 47 (2001) 2045–2049.
E-mail address :jadoming@usal.es, jmp@usal.es and laina@usal.es
Departamento de Matem´
aticas, Universidad de Salamanca, Plaza de la Merced 1-4,
37008 Salamanca, Spain
