Article

Centraliser codes

Authors:
A. Alahmadi at King Abdulaziz University
A. Alahmadi
Shefa Alamoudi
Shefa Alamoudi
Shefa Alamoudi
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Suat Karadeniz at Texas A&M University – Texarkana
Suat Karadeniz
Bahattin Yildiz at Northern Arizona University
Bahattin Yildiz
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Abstract

Centraliser codes are codes of length n2n2 defined as centralisers of a given matrix A of order n. Their dimension, parity-check matrices, syndromes, and automorphism groups are investigated. A lower bound on the dimension is n, the order of A. This bound is met when the minimal polynomial is equal to the annihilator, i.e. for so-called cyclic (a.k.a. non-derogatory) matrices. If, furthermore, the matrix is separable and the adjacency matrix of a graph, the automorphism group of that graph is shown to be abelian and to be even trivial if the alphabet field is of even characteristic.

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... It is easy to see that C F q (A, γ) is a linear subspace of the matrix space F n×n q over F q , and hence C F q (A, γ) is a linear code, called a twisted centralizer code, whose elements (i.e., codewords) are matrices, that can be viewed as vectors of length n 2 , by reading them column-by-column. The notion of twisted centralizer codes is a generalization of centralizer codes [1], since a centralizer code is a twisted centralizer code, twisted by 1 = γ ∈ F q . ...
... Regarding twisted centralizer codes, so far, Alahmadi et al. ( [2], [3]) have determined an upper and lower bound for the dimension, and obtained the exact values of the dimension only for cyclic or diagonalizable matrices A. Moreover, they [1] also state, "In general determining the dimension of such a code 2 given A is a non-trivial problem, and we were only able to give a spectral characterization of the dimension over an extension of the base field". 3 Continuing and improving the results obtained by Alahmadi et al. ...
... Namely, a twisted centralizer code with γ = 1. 3[1], p. 76. ...
On bases and the dimensions of twisted centralizer codes
Article
  • Aug 2021
  • FINITE FIELDS TH APP
Alahmadi et al. (2017) [2] introduced the notion of twisted centralizer codes, CFq(A,γ), defined asCFq(A,γ)={X∈Fqn×n:AX=γXA}, for A∈Fqn×n, and γ∈Fq. Moreover, Alahmadi et al. (2017) [3] also investigated the dimension of such codes and obtained upper and lower bounds for the dimension, and the exact value of the dimension only for cyclic or diagonalizable matrices A. Generalizing and sharpening Alahmadi et al.'s results, in this paper, we determine the exact value of the dimension as well as provide an algorithm to construct an explicit basis of the codes for any given matrix A.
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... If |I| = 1 we write C(A, B) instead of C({A}, {B}). Centralizer codes [1] have the form C(A, A) and twisted centralizer codes [2,3] have the form C(A, αA) where A ∈ F r×s and α ∈ F . Intertwining codes C(A, B) are more general still, so our dimension formula (Theorem 2.8) has particularly wide applicability. ...
... Our construction uses k linearly independent 1 × s row vectors u 1 , . . . , u k which give the first k rows of S ∈ GL(s, F ), and k linearly independent r × 1 column vectors v (1) , . . . , v (k) which give the first k columns of R −1 ∈ GL(r, F ). ...
... Since |F | 3, we may choose γ ∈ F \ {1, 1 − s}. As v (1) , . . . , v (k) are linearly independent, there exists an r × r invertible matrix, which we call R −1 , whose first k columns are v (1) , . . . ...
On the parameters of intertwining codes
Article
Full-text available
  • Sep 2018
Let F be a field and let Fr×s denote the space of r×s matrices over F. Given equinumerous subsets A={Ai∣i∈I}⊆Fr×r and B={Bi∣i∈I}⊆Fs×s we call the subspace C(A,B):={X∈Fr×s∣AiX=XBi for i∈I} an \emph{intertwining code}. We show that if C(A,B)≠{0}, then for each i∈I, the characteristic polynomials of Ai and Bi and share a nontrivial factor. We give an exact formula for k=dim(C(A,B)) and give upper and lower bounds. This generalizes previous work in this area. Finally we construct intertwining codes with large minimum distance when the field is not `too small'. We give examples of codes where d=rs/k=1/R is large where the minimum distance, dimension, and rate of the linear code C(A,B) by d, k, and R=k/rs, respectively.
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Article
Full-text available
  • Jan 2018
The barrier of the family of centralizer codes is the length which is always n². In our paper, we have taken codes generated by two matrices A and C of different orders n × n and k × k respectively. This family of codes are termed as intertwining codes and denoted by . Specialty of this code is the length nk which gives a new approach to characterize family of centralizer codes. In this article, we show an upper bound on the minimum distance of intertwining codes. Besides, we establish two decoding methods which can be fitted to intertwining codes as well as for any linear codes. Moreover, we have shown a condition for which a linear code can be represented as an intertwining code.
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Twisted Centralizer Codes
Article
  • Aug 2016
  • LINEAR ALGEBRA APPL
Given an n×nn\times n matrix A over a field F and a scalar a∈Fa\in F, we consider the linear codes C(A,a):={B∈Fn×n∣ AB=aBA}C(A,a):=\{B\in F^{n\times n}\mid \,AB=aBA\} of length n2n^2. We call C(A,a) a twisted centralizer code. We investigate properties of these codes including their dimensions, minimum distances, parity-check matrices, syndromes, and automorphism groups. The minimal distance of a centralizer code (when a=1) is at most n, however for a≠0,1a\ne 0,1 the minimal distance can be much larger, as large as n2n^2.
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On the dimension of twisted centralizer codes
Article
  • Jul 2016
Given a field F, a scalar λ∈F\lambda\in F and a matrix A∈Fn×nA\in F^{n\times n}, the twisted centralizer code CF(A,λ):={B∈Fn×n∣AB−λBA=0}C_F(A,\lambda):=\{B\in F^{n\times n}\mid AB-\lambda BA=0\} is a linear code of length n2n^2. When A is cyclic and λ≠0\lambda\ne0 we prove that dim⁡CF(A,λ)=deg(gcd⁡(cA(t),λncA(λ−1t)))\dim C_F(A,\lambda)=\mathrm{deg}(\gcd(c_A(t),\lambda^n c_A(\lambda^{-1}t))) where cA(t)c_A(t) denotes the characteristic polynomial of A. We also show how CF(A,λ)C_F(A,\lambda) decomposes, and we estimate the probability that CF(A,λ)C_F(A,\lambda) is nonzero when ∣F∣|F| is finite. Finally, we prove dim⁡CF(A,λ)⩽n2/2\dim C_F(A,\lambda)\leqslant n^2/2 for λ∉{0,1}\lambda\not\in\{0,1\} and `almost all' matrices A.
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Article
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  • Oct 2004
  • LINEAR ALGEBRA APPL
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We show that, if the eigenvalues of the adjacency matrix of a graph are distinct, then the group of automorphisms of the graph is Abelian.
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