Construction of additive complementary dual codes over finite fields

Additive codes have become an increasingly important topic in algebraic coding theory due to their applications in quantum error-correction and quantum computing. Linear Complementary Dual (LCD) codes play an important role for improving the security of information against certain attacks. Motivated by these facts, we define additive complementary dual codes (ACD for short) over a finite abelian group in terms of an arbitrary duality on the ambient space and examine their properties. We show that the best minimum weight of ACD codes is always greater than or equal to the best minimum weight of LCD codes of the same size and that this inequality is often strict. We give some matrix constructions for quaternary ACD codes from a given quaternary ACD code and also from a given binary self-orthogonal code. Moreover, we construct an algorithm to determine if a given quaternary additive code is an ACD code with respect to all possible symmetric dualities. We also determine the largest minimum distance of quaternary ACD codes for lengths <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n \leq 10$ </tex-math></inline-formula> . The obtained codes are either optimal or near optimal according to Bierbrauer <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> +. (2009).

Linear complementary dual codes have become an interesting sub-family of linear codes over finite fields since they can be practically applied in various fields such as cryptography and quantum error-correction. Recently, properties of complementary dual abelian codes were established in group algebras of arbitrary finite abelian groups. However, the enumeration formulas were given mostly based on number-theoretical characteristic functions. In this article, complementary dual abelian codes determined by some finite abelian groups are revisited. Specifically, the characterization of cyclotomic classes of an abelian group and the enumeration of complementary dual abelian codes are presented, where the group is a finite abelian p-group, a finite abelian 2-group, and a product of a finite abelian p-group and a finite abelian 2-group for some odd prime number p different from the characteristic of the alphabet filed. The enumeration formula for such complementary dual codes is given explicitly in a more precise form without characteristic functions. Some illustrative examples are given as well.

Aydogdu et al. studied the standard forms of generator and parity-check matrices of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive codes, and presented generators of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive cyclic codes (Finite Fields Appl. 48: 241-260, 2017). In this paper, we investigate some other useful properties of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive codes, including asymptotically good $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive cyclic codes and $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive complementary dual codes. The present paper can be viewed as a necessary complementary part of Aydogdu’s work.

Multi-twisted additive self-orthogonal and ACD codes are asymptotically good

Theory of additive complementary dual codes, constructions and computations

The additive codes may have better parameters than linear codes. However, it is still a challenging problem to efficiently construct additive codes that outperform linear codes, especially those with greater distances than linear codes of the same lengths and dimensions. This paper focuses on constructing additive codes that outperform linear codes based on quasi-cyclic codes and combinatorial methods. Firstly, we propose a lower bound on the symplectic distance of 1-generator quasi-cyclic codes of index even. Secondly, we get many binary quasi-cyclic codes with large symplectic distances utilizing computer-supported combination and search methods, all of which correspond to good quaternary additive codes. Notably, some additive codes have greater distances than best-known quaternary linear codes in Grassl’s code table (bounds on the minimum distance of quaternary linear codes http://www.codetables.de) for the same lengths and dimensions. Moreover, employing a combinatorial approach, we partially determine the parameters of optimal quaternary additive 3.5-dimensional codes with lengths from 28 to 254. Finally, as an extension, we also construct some good additive complementary dual codes with larger distances than the best-known quaternary linear complementary dual codes in the literature.

In this paper, we introduce a new class of additive codes over finite fields, viz. multi-twisted (MT) additive codes, which are generalizations of constacyclic additive codes. We study their algebraic structures by writing a canonical form decomposition and provide an enumeration formula for these codes. By placing ordinary, Hermitian and $$*$$ trace bilinear forms, we further study their dual codes and derive necessary and sufficient conditions under which a MT additive code is self-dual and self-orthogonal. We also derive a necessary and sufficient condition for the existence of a self-dual MT additive code over a finite field, and provide enumeration formulae for all self-dual and self-orthogonal MT additive codes over finite fields with respect to the aforementioned trace bilinear forms. We also obtain several good codes within the family of MT additive codes over finite fields.

The construction of linear codes from functions in finite fields has been widely studied in the literature. There are two generic construction methods: the first and second generic construction methods for generating linear codes from functions over finite fields. In this paper, we first define the augmented code construction of the variation of the second generic construction method and then present new infinite families of four- and five-weight self-orthogonal divisible codes derived from trace functions. Moreover, by using the augmented code construction based on the first generic construction method, we construct new infinite families of three-weight and four-weight self-orthogonal divisible codes from weakly regular plateaued functions. We determine all parameters of the constructed self-orthogonal codes as well as their dual codes over the odd characteristic finite fields. We present Hamming weights and their weight distributions for the constructed self-orthogonal codes. Additionally, we utilise the constructed p-ary self-orthogonal codes to develop p-ary Linear Complementary Dual (LCD) codes and determine the parameters of the obtained LCD codes and their dual codes.

We determine the number of symmetric dualities on the $ s $-fold cross product of the cyclic group of order $ 2, $ which is the additive group of the finite field $ {\mathbb{F}}_{2^s}. $ We show that the ratio of symmetric dualities over all dualities goes to $ 0 $ as $ s $ goes to infinity.We also prove a surprising result that given any two binary codes $ C $ and $ D $ of the same length $ n $ with $ |C||D| = 2^n $, then viewing them as groups there is a symmetric duality $ M $ with $ C^M = D $, which also relates their weight enumerators as additive codes in a group via the MacWilliams relations. Using this theorem we show that any additive code in this setting can be viewed as an additive complementary dual code of length $ 1 $ with respect to some duality.

Abelian codes and complementary dual codes form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, a family of abelian codes with complementary dual in a group algebra $${\mathbb {F}}_{p^\nu }[G]$$ has been studied under both the Euclidean and Hermitian inner products, where p is a prime, $$\nu $$ is a positive integer and G is an arbitrary finite abelian group. Based on the discrete Fourier transform decomposition for semi-simple group algebras and properties of ideas in local group algebras, the characterization of such codes have been given. Subsequently, the number of complementary dual abelian codes in $${\mathbb {F}}_{p^\nu }[G]$$ has been shown to be independent of the Sylow p-subgroup of G and it has been completely determined for every finite abelian group G. In some cases, a simplified formula for the enumeration has been provided as well. The known results for cyclic complementary dual codes can be viewed as corollaries.

Quasi-cyclic complementary dual codes

Self-conjugate-reciprocal irreducible monic factors of xn − 1 over finite fields and their applications

This review article discusses linear complementary duals over finite fields as counter-measures to side-channel attacks, faulty injection attacks, hardware trojan horse attacks, etc. We first discuss some security attacks and the need for linear complementary codes. Then introduce the linear complementary dual codes and study their properties. Some constructions and bounds on the linear complementary codes over finite fields are emphasised. Linear complementary dual codes with respect to different inner products such as Hermitian, Galois, Sigma, etc., are considered in our discussion. We end the survey with some comments on codes over rings.

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes have been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over Fq 2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. This paper also derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum Bose-Chaudhuri-Hocquenghem (BCH) codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper

In this paper we define and study rank metric codes endowed with a Hermitian form. We analyze the duality for $\mathbb {F}_{q^{2}}$ -linear matrix codes in the ambient space $(\mathbb {F}_{q^{2}})_{n,m}$ and for both $\mathbb {F}_{q^{2}}$ -additive codes and $\mathbb {F}_{q^{2m}}$ -linear codes in the ambient space $\mathbb {F}_{q^{2m}}^{n}$ . Similarly, as in the Euclidean case we establish a relationship between the duality of these families of codes. For this we introduce the concept of $q^{m}$ -duality between bases of $\mathbb {F}_{q^{2m}}$ over $\mathbb {F}_{q^{2}}$ and prove that a $q^{m}$ -self dual basis exists if and only if $m$ is an odd integer. We obtain connections on the dual codes in $\mathbb {F}_{q^{2m}}^{n}$ and $(\mathbb {F}_{q^{2}})_{n,m}$ with the corresponding inner products. In particular, we study Hermitian linear complementary dual, Hermitian self-dual and Hermitian self-orthogonal codes in $\mathbb {F}_{q^{2m}}^{n}$ and $(\mathbb {F}_{q^{2}})_{n,m}$ . Furthermore, we present connections between Hermitian $\mathbb {F}_{q^{2}}$ -additive codes and Euclidean $\mathbb {F}_{q^{2}}$ -additive codes in $\mathbb {F}_{q^{2m}}^{n}$ .