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Shape enumerators of self-dual NRT codes over finite fields

Yin Chen, Runxuan Zhang

Abstract

We use invariant theory of finite groups to study shape enumerators of self-dual linear codes in a finite NRT metric space. We provide a new approach that avoids applying Molien's formula to compute all possible shape enumerators. We also explicitly compute the shape enumerators of some low-dimensional self-dual NRT codes over an arbitrary finite field.

Shape enumerators of self-dual NRT codes over finite fields

Abstract

We use invariant theory of finite groups to study shape enumerators of self-dual linear codes in a finite NRT metric space. We provide a new approach that avoids applying Molien's formula to compute all possible shape enumerators. We also explicitly compute the shape enumerators of some low-dimensional self-dual NRT codes over an arbitrary finite field.
Paper Structure (10 sections, 5 theorems, 63 equations)

This paper contains 10 sections, 5 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. NRT Codes over Finite Fields
  3. Finite NRT spaces
  4. Finite NRT codes
  5. Shape enumerators
  6. Representations and Polynomial Invariants of $C_2$
  7. Representation theory of $C_2$
  8. Polynomial invariants of $C_2$
  9. An algorithm
  10. Examples

Key Result

Lemma 2.1

The trace of $q^{-\frac{m}{2}}\cdot g$ is $1$ when $m$ is even.

Theorems & Definitions (17)

  • Lemma 2.1
  • proof
  • Proposition 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • proof : Proof of Proposition \ref{['prop3.1']}
  • Remark 3.4
  • proof
  • ...and 7 more