2,105 research outputs found
On the weight enumerator of product codes
AbstractThe number of words of weight w in the product code of linear codes with minimum distances dr and dc is expressed in the number of low weight words of the constituent codes, provided that w < drdc + max(dr, dc). By examples it is shown that, in general, the full weight enumerator of a product code is not completely determined by the weight enumerator of its constituent codes
Zeta Functions for Tensor Codes
In this work we introduce a new class of optimal tensor codes related to the
Ravagnani-type anticodes, namely the -tensor maximum rank distance codes. We
show that it extends the family of -maximum rank distance codes and contains
the -tensor binomial moment determined codes (with respect to the
Ravagnani-type anticodes) as a proper subclass. We define and study the
generalized zeta function for tensor codes. We establish connections between
this object and the weight enumerator of a code with respect to the
Ravagnani-type anticodes. We introduce a new refinement of the invariants of
tensor codes exploiting the structure of product lattices of some classes of
anticodes and we derive the corresponding MacWilliams identities. In this
framework, we also define a multivariate version of the tensor weight
enumerator and we establish relations with the corresponding zeta function. As
an application we derive connections on the generalized tensor weights related
to the Delsarte and Ravagnani-type anticodes
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
Fourier-Reflexive Partitions and MacWilliams Identities for Additive Codes
A partition of a finite abelian group gives rise to a dual partition on the
character group via the Fourier transform. Properties of the dual partitions
are investigated and a convenient test is given for the case that the bidual
partition coincides the primal partition. Such partitions permit MacWilliams
identities for the partition enumerators of additive codes. It is shown that
dualization commutes with taking products and symmetrized products of
partitions on cartesian powers of the given group. After translating the
results to Frobenius rings, which are identified with their character module,
the approach is applied to partitions that arise from poset structures
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