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The MacWilliams Identity for the Hermitian Rank Metric

Izzy Friedlander

TL;DR

This paper extends the MacWilliams identity to codes based on Hermitian matrices under the rank metric by introducing a negative-$q$ algebra. It develops a complete transform framework, including the negative-$q$-product, negative-$q$-transform, and negative-$q$ derivatives, to relate the weight distributions of a code and its dual via a q-analog functional transform. Central to the approach are the Negative-$q$-Krawtchouk polynomials, identified with the eigenvalues of the Hermitian association scheme, which underpin the MacWilliams identity for the Hermitian rank metric. The results yield explicit weight-distribution formulas and moment relations, including for maximal Hermitian rank-distance (MHRD) codes, and provide a path toward generalizing to other translation schemes in association schemes literature.

Abstract

Error-correcting codes have an important role in data storage and transmission and in cryptography, particularly in the post-quantum era. Hermitian matrices over finite fields and equipped with the rank metric have the potential to offer enhanced security with greater efficiency in encryption and decryption. One crucial tool for evaluating the error-correcting capabilities of a code is its weight distribution and the MacWilliams Theorem has long been used to identify this structure of new codes from their known duals. Earlier papers have developed the MacWilliams Theorem for certain classes of matrices in the form of a functional transformation, developed using $q$-algebra, character theory and Generalised Krawtchouk polynomials, which is easy to apply and also allows for moments of the weight distribution to be found. In this paper, recent work by Kai-Uwe Schmidt on the properties of codes based on Hermitian matrices such as bounds on their size and the eigenvalues of their association scheme is extended by introducing a negative-$q$ algebra to establish a MacWilliams Theorem in this form together with some of its associated moments. The similarities in this approach and in the paper for the Skew-Rank metric by Friedlander et al. have been emphasised to facilitate future generalisation to any translation scheme.

The MacWilliams Identity for the Hermitian Rank Metric

TL;DR

This paper extends the MacWilliams identity to codes based on Hermitian matrices under the rank metric by introducing a negative- algebra. It develops a complete transform framework, including the negative--product, negative--transform, and negative- derivatives, to relate the weight distributions of a code and its dual via a q-analog functional transform. Central to the approach are the Negative--Krawtchouk polynomials, identified with the eigenvalues of the Hermitian association scheme, which underpin the MacWilliams identity for the Hermitian rank metric. The results yield explicit weight-distribution formulas and moment relations, including for maximal Hermitian rank-distance (MHRD) codes, and provide a path toward generalizing to other translation schemes in association schemes literature.

Abstract

Error-correcting codes have an important role in data storage and transmission and in cryptography, particularly in the post-quantum era. Hermitian matrices over finite fields and equipped with the rank metric have the potential to offer enhanced security with greater efficiency in encryption and decryption. One crucial tool for evaluating the error-correcting capabilities of a code is its weight distribution and the MacWilliams Theorem has long been used to identify this structure of new codes from their known duals. Earlier papers have developed the MacWilliams Theorem for certain classes of matrices in the form of a functional transformation, developed using -algebra, character theory and Generalised Krawtchouk polynomials, which is easy to apply and also allows for moments of the weight distribution to be found. In this paper, recent work by Kai-Uwe Schmidt on the properties of codes based on Hermitian matrices such as bounds on their size and the eigenvalues of their association scheme is extended by introducing a negative- algebra to establish a MacWilliams Theorem in this form together with some of its associated moments. The similarities in this approach and in the paper for the Skew-Rank metric by Friedlander et al. have been emphasised to facilitate future generalisation to any translation scheme.
Paper Structure (22 sections, 29 theorems, 151 equations, 1 table)

This paper contains 22 sections, 29 theorems, 151 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hermitian Matrices
  4. Codes based on Subspaces of Hermitian Matrices
  5. Counting the number of Hermitian Matrices of a given size
  6. Inner product of two Hermitian Matrices
  7. Negative-$q$ Gaussian Coefficients and other useful identities
  8. The Negative-$q$-Product and Negative-$q$-Transform
  9. The Negative-$q$-product, Negative-$q$-power and the Negative-$q$-transform
  10. Using the Negative-$q$-product to identify the Rank Weight Enumerator of Hermitian Matrices
  11. The MacWilliams Identity for the Hermitian Rank Metric
  12. Eigenvalues of the Association Scheme of Hermitian Matrices
  13. The Negative-$q$-Krawtchouk Polynomial
  14. The MacWilliams Identity for the Hermitian Rank Metric
  15. The Negative-$q$-Derivatives
  16. ...and 7 more sections

Key Result

Theorem 2.2

$\mathscr{H}_{q,t}$ is a $t^2$-dimensional vector space over $\mathbb{F}_{q}$.

Theorems & Definitions (72)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Theorem 2.6
  • Example 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 62 more