Weight distribution of rank-metric codes

Cruz, Javier de la; Gorla, Elisa; López, Hiram H.; Ravagnani, Alberto

doi:10.1007/s10623-016-0325-1

Cited by 27 publications

(16 citation statements)

References 8 publications

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“…7 We have relations between B (i) u s and B (i) u s. For the family of closure-type anticodes, these quantities are not equal in general. 8 Extended in Proposition 5.3 9 Extended in Proposition 5.10…”

Section: Appendixmentioning

confidence: 99%

Tensor Codes and their Invariants

Byrne¹,

Cotardo²

2021

Preprint

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In 1991, Roth introduced a natural generalization of rank metric codes, namely tensor codes. The latter are defined to be subspaces of r-tensors where the ambient space is endowed with the tensor rank as a distance function. In this work, we describe the general class of tensor codes and we study their invariants that correspond to different families of anticodes. In our context, an anticode is a perfect space that has some additional properties. A perfect space is one that is spanned by tensors of rank 1. Our use of the anticode concept is motivated by an interest in capturing structural properties of tensor codes. In particular, we indentify four different classes of tensor anticodes and show how these gives different information on the codes they describe. We also define the generalized tensor binomial moments and the generalized tensor weight distribution of a code and establish a bijection between these invariants. We use the generalized tensor binomial moments to define the concept of an i-tensor BMD code, which is an extremal code in relation to an inequality arising from them. Finally, we give MacWilliams identities for generalized tensor binomial moments.

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Section: Appendixmentioning

confidence: 99%

Tensor Codes and their Invariants

Byrne¹,

Cotardo²

2021

Preprint

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“…(ii) According to Lemma 4.3 b) we have |D d (C)| = A d (C) q m −1 . Since A d+1 (C) = 0, Theorem 25 of [5] yields…”

Section: Q-analog Steiner Systems and Rank Metric Codesmentioning

confidence: 99%

On q-Steiner systems from rank metric codes

Arias

Cruz

Rosenthal

et al. 2018

Discrete Mathematics

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“…Else, they are a different family of codes, called quasi MRD codes. This last family of codes was introduced and studied in [2].…”

Section: Introductionmentioning

confidence: 99%

Quasi optimal anticodes: structure and invariants

Gorla¹,

Cristina²

2022

Preprint

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It is well-known that the dimension of optimal anticodes in the rank-metric is divisible by the maximum m between the number of rows and columns of the matrices. Moreover, for a fixed k divisible by m, optimal rank-metric anticodes are the codes with least maximum rank, among those of dimension k. In this paper, we study the family of rank-metric codes whose dimension is not divisible by m and whose maximum rank is the least possible for codes of that dimension, according to the Anticode bound. As these are not optimal anticodes, we call them quasi optimal anticodes (qOACs). In addition, we call dually qOAC a qOAC whose dual is also a qOAC. We describe explicitly the structure of dually qOACs and compute their weight distributions, generalized weights, and associated q-polymatroids.

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Weight distribution of rank-metric codes

Cited by 27 publications

References 8 publications

Tensor Codes and their Invariants

Tensor Codes and their Invariants

On q-Steiner systems from rank metric codes

Quasi optimal anticodes: structure and invariants

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