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MacWilliams duality for rank metric codes over finite chain rings

Iván Blanco-Chacón, Alberto F. Boix, Marcus Greferath, Erik Hieta-aho

TL;DR

This work generalizes Ravagnani's MacWilliams duality for rank-metric codes from finite fields to finite chain rings by leveraging a non-degenerate trace bilinear form and a shortening framework. It derives a coefficient-wise MacWilliams identity that links the $q$-binomial moments of a rank-metric code over a chain ring with those of its dual, and proves that the dual of an MRD code is also MRD. Key steps include a detailed treatment of shortening by free submodules, a cardinality equality $|M|=|M^*|$ for modules over chain rings, and a central binomial-moment formula that governs the relationship between the rank distributions of a code and its dual. The results broaden the applicability of rank-metric coding theory to ring alphabets and establish foundational duality results, with open questions about MacWilliams transforms and zeta-function formulations for these codes.

Abstract

We extend Ravagnani's MacWilliams duality theory to the settings of rank metric codes over finite chain rings, relating the sequences of $q$-binomial moments of a rank metric code over this class of rings with those of its dual.

MacWilliams duality for rank metric codes over finite chain rings

TL;DR

This work generalizes Ravagnani's MacWilliams duality for rank-metric codes from finite fields to finite chain rings by leveraging a non-degenerate trace bilinear form and a shortening framework. It derives a coefficient-wise MacWilliams identity that links the -binomial moments of a rank-metric code over a chain ring with those of its dual, and proves that the dual of an MRD code is also MRD. Key steps include a detailed treatment of shortening by free submodules, a cardinality equality for modules over chain rings, and a central binomial-moment formula that governs the relationship between the rank distributions of a code and its dual. The results broaden the applicability of rank-metric coding theory to ring alphabets and establish foundational duality results, with open questions about MacWilliams transforms and zeta-function formulations for these codes.

Abstract

We extend Ravagnani's MacWilliams duality theory to the settings of rank metric codes over finite chain rings, relating the sequences of -binomial moments of a rank metric code over this class of rings with those of its dual.
Paper Structure (7 sections, 19 theorems, 56 equations)

This paper contains 7 sections, 19 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on modules over finite chain rings
  3. MacWilliams duality theory
  4. Shortening and strict shortening
  5. Cardinality of a module and its dual
  6. Binomial moments of the rank distributions of a code and its dual
  7. Conclusions

Key Result

Proposition 2.6

rankmetriccodesoverpir. Let $R$ be a chain ring, and let $A\in M_{m,n} (R).$ Then, there is an $R$--module isomorphism $\operatorname{CS}(A)\cong\operatorname{RS}(A).$ Moreover, we have $\operatorname{rank}(A)=\mu_R(\operatorname{RS}(A)).$

Theorems & Definitions (48)

  • Definition 2.1: Finite chain rings
  • Definition 2.2: Galois Ring
  • Definition 2.3: Code over a finite chain ring
  • Definition 2.4: The rank metric
  • Example 2.5
  • Proposition 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • Corollary 2.10
  • ...and 38 more