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Affine-invariant extended cyclic codes and partially ordered sets of antichains
Abstract
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Let p be a prime and let be positive integers such that r|e and e|m. Extended cyclic codes of length pm over Fpr which are invariant under are characterized by a well-known relation ≪e on the set {0,1,…,pm-1}. From the relation ≪e, we derive a partial order ≺ in defined by an e-dimensional simplicial cone. We show that the aforementioned extended cyclic codes can be enumerated by the ideals of (U,≺) which are invariant under the rth power of a circulant permutation matrix. When e=2, we enumerate all such invariant ideals by describing their boundaries. Explicit formulas are obtained for the total number of -invariant extended cyclic codes of length pm over Fpr and for the dimensions of such codes. We also enumerate all self-dual -invariant extended cyclic codes of length 2m over F22 where is odd; the restrictions on the parameters are necessary conditions for the existence of self-dual affine invariant extended cyclic codes with e=2.
Affine-invariant codes are extended cyclic codes of length p
m
invariant under the affine-group acting on
. This class of codes includes codes of great interest such as extended narrow-sense BCH codes. In recent papers, we classified the automorphism groups of affine-invariant codes berg, bech1. We derive here new results, especially when the alphabet field is an extension field, by expanding our previous tools. In particular we complete our results on BCH codes, giving the automorphism groups of extended narrow-sense BCH codes defined over any extension field.
A ( v, k, λ)-difference set D in a group G can be used to create a symmetric 2-( v, k, λ) design,
, from which arises a code C, generated by vectors corresponding to the characteristic function of blocks of
. This paper examines properties of the code C, and of a subcode, C
o=JC, where J is the radical of the group algebra of G over
. When G is a 2-group, it is shown that Co is equivalent to the first-order Reed-Muller code,
, precisely when the 2-divisor of Co is maximal. In addition, ifD is a non-trivial difference set in an elementary abelian 2-group, and if D is generated by a quadratic bent function, then Co is equal to a power of the radical. Finally, an example is given of a difference set whose characteristic function is not quadratic, although the 2-divisor of Co is maximal.
LetC be an extended cyclic code of lengthp
m
over
. The border ofC is the set of minimal elements (according to a partial order on [0,p
m
−1]) of the complement of the defining-set ofC. We show that an affine-invariant code whose border consists of only one cyclotomic coset is the dual of an extended BCH code if, and only if, this border is the cyclotomic coset, sayF(t, i), ofp
t
−1−i, with 1 ≦t ≦ m and 0 ≦i < p−1. We then study such privileged codes. We first make precize which duals of extendedBCH codes they are. Next, we show that Weil's bound in this context gives an explicit formula; that is, the couple (t, i) fully determines the value of the Weil bound for the code with borderF(t, i). In the case where this value is negative, we use the Roos method to bound the minimum distance, greatly improving the BCH bound.
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