{"title":"On dual-containing, almost dual-containing matrix-product codes and related quantum codes","authors":"Meng Cao","doi":"10.1016/j.ffa.2024.102400","DOIUrl":null,"url":null,"abstract":"<div><p>Matrix-product (MP) codes are a type of long codes formed by combining several commensurate constituent codes with a defining matrix. In this paper, we study the MP code when the defining matrix <em>A</em> satisfies the condition that <span><math><mi>A</mi><msup><mrow><mi>A</mi></mrow><mrow><mo>⊤</mo></mrow></msup></math></span> is <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>-monomial. We give an explicit formula for calculating the dimension of the hull of a MP code. We present the necessary and sufficient conditions for a MP code to be dual-containing (DC), almost dual-containing (ADC), self-orthogonal (SO) and almost self-orthogonal (ASO), respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code that is DC, ADC, SO and ASO, respectively. We give alternative necessary and sufficient conditions for a MP code to be ADC and ASO, respectively, and show several cases where a MP code is not ADC or ASO. We give the construction methods of DC and ADC MP codes, including those with optimal minimum distance lower bounds. We introduce the notation of <em>τ</em>-optimal defining (<em>τ</em>-OD) matrices and provide the criteria for determining whether two types of <span><math><mi>k</mi><mo>×</mo><mi>k</mi></math></span> matrices are <em>τ</em>-OD matrices at <span><math><mi>k</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><mi>k</mi><mo>=</mo><mn>4</mn></math></span>, respectively. We give many examples of DC and ADC MP codes involving <em>τ</em>-OD matrices, some of which are optimal or almost optimal according to the Database <span>[11]</span>. By applying the generalized Steane's enlargement procedure to these DC MP codes, we obtain some good quantum codes that improve those available in the Database <span>[7]</span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S107157972400039X","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Matrix-product (MP) codes are a type of long codes formed by combining several commensurate constituent codes with a defining matrix. In this paper, we study the MP code when the defining matrix A satisfies the condition that is -monomial. We give an explicit formula for calculating the dimension of the hull of a MP code. We present the necessary and sufficient conditions for a MP code to be dual-containing (DC), almost dual-containing (ADC), self-orthogonal (SO) and almost self-orthogonal (ASO), respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code that is DC, ADC, SO and ASO, respectively. We give alternative necessary and sufficient conditions for a MP code to be ADC and ASO, respectively, and show several cases where a MP code is not ADC or ASO. We give the construction methods of DC and ADC MP codes, including those with optimal minimum distance lower bounds. We introduce the notation of τ-optimal defining (τ-OD) matrices and provide the criteria for determining whether two types of matrices are τ-OD matrices at and , respectively. We give many examples of DC and ADC MP codes involving τ-OD matrices, some of which are optimal or almost optimal according to the Database [11]. By applying the generalized Steane's enlargement procedure to these DC MP codes, we obtain some good quantum codes that improve those available in the Database [7].
期刊介绍:
Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering.
For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods.
The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.