LCD codes from tridiagonal Toeplitz matrices

Abstract

Double Toeplitz (DT) codes are codes with a generator matrix of the form $(I,T)$ with T a Toeplitz matrix, that is to say constant on the diagonals parallel to the main. When T is tridiagonal and symmetric we determine its spectrum explicitly by using Dickson polynomials, and deduce from there conditions for the code to be LCD. Using a special concatenation process, we construct optimal or quasi-optimal examples of binary and ternary LCD codes from DT codes over extension fields.

Introduction

Linear Complementary Dual (LCD) codes are linear codes which intersect their dual trivially. They were introduced by Massey in 1992 to solve a problem in Information Theory [10]. They were proved to be asymptotically good by Sendrier [11], who used them in relation with equivalence testing of linear codes [12]. They enjoyed a renewal of interest in 2016, with an application to side-channel attacks on embarked cryptosystems [5]. Recently LCD double circulant codes or double negacirculant codes were constructed over various alphabets [8], [13], [14], [15], [17]. A far reaching generalization of both double circulant and double negacirculant codes is that of double Toeplitz codes [16]. In the present paper, we introduce a class of double Toeplitz codes which can be effectively tested for being LCD.

A code is double Toeplitz (DT) if its generator matrix is of the form $(I,T)$ with I an identity matrix, and T a Toeplitz matrix of the same order. Recall that a matrix is Toeplitz if it has constant entries on all diagonals parallel to the main diagonal. Thus circulant matrices and negacirculant matrices are Toeplitz.

It is easy to check that such a code is LCD iff −1 is not an eigenvalue of $T{T}^t$. To make that condition easy to check we will make two hypotheses on T:

• $T=T^t$ implying $T{T}^t=T^2$;

• T is tridiagonal, in the sense that $T_{ij}=0$ if $|i-j|>1$.

In the next section, we show that the characteristic polynomial of a tridiagonal symmetric Toeplitz matrix satisfies a three-term recurrence that can be identified, up to an easy change of variable to that of the Dickson polynomials [9]. The roots of these polynomials can be determined explicitly [3]. Hence we obtain an exact and explicit characterization on whether a given DT code $(I,T)$ is LCD or not, when T is tridiagonal and symmetric (see Theorem 2.9, Theorem 2.10 below). It seems very difficult to obtain such a characterization for arbitrary Toeplitz T. Moreover this is the first paper in the literature using factorization of Dickson polynomials for the characterization of some LCD codes as far as we know.

Under some mild arithmetic conditions we can show that this spectrum does not intersect the base field, and in particular does not contain −1. Some sufficient conditions for the DT code to be LCD follow. Since the DT codes so constructed have minimum distance at most three, a rather sophisticated concatenation process, namely isometry (see Definition 3.1 below) can be used to construct an LCD code over a small field. Note that because of the fundamental result that any linear code over ${\mathbb{F}}_q$ with $q>3$ is equivalent to an LCD code [6], the theory of LCD codes is focusing on the cases of ${\mathbb{F}}_2$ and ${\mathbb{F}}_3$. Using the said concatenation process optimal or quasi-optimal LCD codes over these two fields are explicitly constructed.

The material is organized as follows. The next section studies the spectrum of Toeplitz matrices. Section 3 describes a concatenation process that allow for LCD codes over small fields. Numerical examples are given there. The last section concludes the paper.