Refinements of Levenshtein Bounds in q-ary Hamming Spaces

Refinements of Levenshtein bounds in q-ary Hamming spaces

We develop refinements of the Levenshtein bound in q-ary Hamming spaces by taking into account the discrete nature of the distances versus the continuous behavior of certain parameters used by Levenshtein. The first relevant cases are investigated in detail and new bounds are presented. In particular, we derive generalizations and q-ary analogs of a MacEliece bound. We provide evidence that our...

On permutation automorphism groups of q-ary Hamming codes

It is established that for any q > 2 the permutation automorphism group of a q-ary Hamming code of length n = (q − 1)/(q− 1) is isomorphic to the unitriangular group UTm(q).

Relative generalized Hamming weights of $q$-ary Reed-Muller codes

Coset constructions of q-ary Reed-Muller codes can be used to store secrets on a distributed storage system in such a way that only parties with access to a large part of the system can obtain information while still allowing for local error-correction. In this paper we determine the relative generalized Hamming weights of these codes which can be translated into a detailed description of the i...

Generalized Hamming Weights of q-ary Reed-Muller Codes

The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as the q-ary Reed-Muller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition.

Improved bounds on identifying codes in binary Hamming spaces