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On codes induced from Hadamard matrices

Ted Hurley

TL;DR

The paper develops unit-derived techniques for Hadamard matrices to systematically construct linear block and convolutional codes with prescribed length, rate, and type, including LCD, self-dual, dual-containing, and quantum error-correcting codes. It introduces core Propositions and Algorithms that, from a single Hadamard matrix, generate large families of codes and analyze their distances, often enabling algebraic distance computation. Extensive explicit examples across various Hadamard sizes and finite fields illustrate LCD, self-dual, DC, and QECC constructions, as well as higher-memory convolutional variants. The discussion of ternary codes highlights rank constraints and specialized constructions over GF(3) and GF(3^2), broadening the applicability to codes over different characteristics. Computational tools (e.g., GAP) are highlighted as valuable for constructing and verifying these codes and distances.

Abstract

Unit derived schemes applied to Hadamard matrices are used to construct and analyse linear block and convolutional codes. Codes are constructed to prescribed types, lengths and rates and multiple series of self-dual, dual-containing, linear complementary dual and quantum error-correcting of both linear block {\em and} convolutional codes are derived.

On codes induced from Hadamard matrices

TL;DR

The paper develops unit-derived techniques for Hadamard matrices to systematically construct linear block and convolutional codes with prescribed length, rate, and type, including LCD, self-dual, dual-containing, and quantum error-correcting codes. It introduces core Propositions and Algorithms that, from a single Hadamard matrix, generate large families of codes and analyze their distances, often enabling algebraic distance computation. Extensive explicit examples across various Hadamard sizes and finite fields illustrate LCD, self-dual, DC, and QECC constructions, as well as higher-memory convolutional variants. The discussion of ternary codes highlights rank constraints and specialized constructions over GF(3) and GF(3^2), broadening the applicability to codes over different characteristics. Computational tools (e.g., GAP) are highlighted as valuable for constructing and verifying these codes and distances.

Abstract

Unit derived schemes applied to Hadamard matrices are used to construct and analyse linear block and convolutional codes. Codes are constructed to prescribed types, lengths and rates and multiple series of self-dual, dual-containing, linear complementary dual and quantum error-correcting of both linear block {\em and} convolutional codes are derived.

Paper Structure

This paper contains 6 sections, 13 theorems, 1 equation, 4 algorithms.

Table of Contents

  1. Introduction
  2. Explicit samples of applications
  3. Additional notation and background
  4. Linear block and convolutional codes induced from Hadamard matrices
  5. Applications/examples
  6. Ternary codes from Hadamard matrices

Key Result

Proposition 2.1

Let $H$ be a Hadamard matrix of size $n$ and $n\neq 0$ in a field $\mathcal{F}$. Suppose $H$ has the form $H= ()$, where $A$ has size $r\times n$, implying $() ()=nI_n$. Then the code generated by $A$ over $\mathcal{F}$ is an LCD $[n,r]$ code $\mathcal{A}$ and $B$ generates the dual code of $\mathca

Theorems & Definitions (27)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 17 more