Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On general self-orthogonal matrix-product codes associated with Toeplitz matrices

Yang Li, Shixin Zhu, Edgar Martínez-Moro

Abstract

In this paper, we present four constructions of {general} self-orthogonal matrix-product codes associated with Toeplitz matrices. The first one relies on the {dual} of a known {general} dual-containing matrix-product code; the second one is founded on {a specific family of} matrices, where we provide an efficient algorithm for generating them {on the basis of Toeplitz matrices} and {it has an interesting application in producing new non-singular by columns quasi-unitary matrices}; and the last two ones are based on the utilization of certain special Toeplitz matrices. Concrete examples and detailed comparisons are provided. As a byproduct, we also find an application of Toeplitz matrices, which is closely related to the constructions of quantum codes.

On general self-orthogonal matrix-product codes associated with Toeplitz matrices

Abstract

In this paper, we present four constructions of {general} self-orthogonal matrix-product codes associated with Toeplitz matrices. The first one relies on the {dual} of a known {general} dual-containing matrix-product code; the second one is founded on {a specific family of} matrices, where we provide an efficient algorithm for generating them {on the basis of Toeplitz matrices} and {it has an interesting application in producing new non-singular by columns quasi-unitary matrices}; and the last two ones are based on the utilization of certain special Toeplitz matrices. Concrete examples and detailed comparisons are provided. As a byproduct, we also find an application of Toeplitz matrices, which is closely related to the constructions of quantum codes.
Paper Structure (11 sections, 16 theorems, 37 equations, 4 tables, 1 algorithm)

This paper contains 11 sections, 16 theorems, 37 equations, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\sigma$ inner products and $\sigma$ duals
  4. Toeplitz matrices and non-singular by columns matrices
  5. Matrix-product codes and their $\sigma$ duals
  6. Four general constructions of $\sigma$ self-orthogonal matrix-product codes associated with Toeplitz matrices
  7. The first general construction via $\sigma'$ duals of known $\sigma'$ dual-containing matrix-product codes
  8. The second general construction via quasi-$\widehat{\sigma}$ matrices obtained from general Toeplitz matrices
  9. The third and fourth general constructions via special Toeplitz matrices
  10. Comparisons of the four general constructions
  11. Concluding remarks

Key Result

Lemma 2.1

( CYW2023) Let $q=p^h$ be a prime power and $e$ be an integer with $0\leq e\leq h-1$. Let ${\mathcal{C}}$ be an $[n,k]_{q}$ linear code. If $\sigma=(\tau, \pi_e)\in \mathbf{SLAut}(\mathbb{F}_q^{n})$, where $\tau$ corresponds to a monomial matrix $M_{\tau}\in \mathcal{M}(\mathbb{F}_q,n)$, then $\sig

Theorems & Definitions (44)

  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • proof
  • Definition 2.8
  • ...and 34 more