Galois LCD codes over finite fields

Linear complementary dual (LCD) codes is a class of linear codes introduced by Massey in 1964. LCD codes have been extensively studied in literature recently. In addition to their applications in data storage, communications systems, and consumer electronics, LCD codes have been employed in cryptography. More specifically, it has been shown that LCD codes can also help improve the security of the information processed by sensitive devices, especially against so-called side-channel attacks (SCA) and fault non-invasive attacks. In this paper, we are interested in the construction of particular algebraic geometry (AG) LCD codes which could be good candidates to be resistant against SCA. We firstly provide a construction scheme for obtaining LCD codes from elliptic curves. Then, some explicit LCD codes from elliptic curve are presented. MDS codes are of the most importance in coding theory due to their theoretical significance and practical interests. In this paper, all the constructed LCD codes from elliptic curves are MDS or almost MDS. Some infinite classes of LCD codes from elliptic curves are optimal due to the Griesmer bound. Finally, we introduce a construction mechanism for obtaining LCD codes from any algebraic curve and derive some explicit LCD codes from hyperelliptic curves and Hermitian curves.

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.

Hermitian linear complementary dual codes are linear codes whose intersections with their Hermitian dual codes are trivial. The largest minimum weight among quaternary Hermitian linear complementary dual codes of dimension $2$ is known for each length. We give the complete classification of optimal quaternary Hermitian linear complementary dual codes of dimension $2$. Hermitian linear complementary dual codes are linear codes whose intersections with their Hermitian dual codes are trivial. The largest minimum weight among quaternary Hermitian linear complementary dual codes of dimension $2$ is known for each length. We give the complete classification of optimal quaternary Hermitian linear complementary dual codes of dimension $2$.

Quasi-cyclic complementary dual codes

Further results on Euclidean and Hermitian linear complementary dual codes

Massey introduced a class of linear code which is called linear complementary dual (LCD) codes. It has been known that LCD codes play a significant role in improving the security of the information processed by sensitive device. In this paper, we constructed 166 quaternary Hermitian linear complementary dual codes with small distance, where 154 codes are new constructions. Besides, it includes 145 Hermitian optimal LCD codes and 9 Hermitian near optimal LCD codes.

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.

An $[{n}, {k}, {n}-{k}+1]$ linear code is called an MDS code. An $[{n}, {k}, {n}-{k}]$ linear code is said to be almost maximum distance separable (almost MDS or AMDS for short). A code is said to be near maximum distance separable (near MDS or NMDS for short) if the code and its dual code both are almost maximum distance separable. The first near MDS code was the [11, 6, 5] ternary Golay code discovered in 1949 by Golay. This ternary code holds 4-designs, and its extended code holds a Steiner system ${S}(5, 6, 12)$ with the largest strength known. In the past 70 years, sporadic near MDS codes holding t-designs were discovered and a lot of infinite families of near MDS codes over finite fields were constructed. However, the question as to whether there is an infinite family of near MDS codes holding an infinite family of t-designs for $ {t}\geq 2$ remains open for 70 years. This paper settles this long-standing problem by presenting an infinite family of near MDS codes over $ {GF}(3^{ {s}})$ holding an infinite family of 3-designs and an infinite family of near MDS codes over $ {GF}(2^{2 {s}})$ holding an infinite family of 2-designs. The subfield subcodes of these two families of codes are also studied, and are shown to be dimension-optimal or distance-optimal.

A linear complementary-dual (LCD) code C is a linear code whose dual code $$C^{\perp }$$ satisfies $$C \cap C^{\perp }=\{0\}$$ . In this work we characterize some classes of LCD q-ary $$(\lambda , l)$$ -quasi-twisted (QT) codes of length $$n=ml$$ with $$(m,q)=1$$ , $$\lambda \in F_{q} \setminus \{0\}$$ and $$\lambda \ne \lambda ^{-1}$$ . We show that every $$(\lambda ,l)$$ -QT code C of length $$n=ml$$ with $$dim(C)<m$$ or $$dim(C^{\perp })<m$$ is an LCD code. A sufficient condition for r-generator QT codes is provided under which they are LCD. We show that every maximal 1-generator $$(\lambda ,l)$$ -QT code of length $$n=ml$$ with $$l>2$$ is either an LCD code or a self-orthogonal code and a sufficient condition for this family of codes is given under which such a code C is LCD. Also it is shown that every maximal 1-generator $$(\lambda ,2)$$ -QT code is LCD. Several good and optimal LCD QT codes are presented.

Linear codes that meet their dual trivially are also known as linear complementary dual codes. Quasi-abelian complementary dual codes are characterized using a known decomposition of a semisimple group algebra. Consequently, enumeration of such codes are obtained. More explicit formulas are given for the number of quasi-abelian complementary dual codes of index 2 with respect to Euclidean and Hermitian inner products. A sequence of asymptotically good binary quasi-abelian complementary dual codes of index 3 is constructed from an existing sequence of asymptotically good binary self-dual quasi-abelian codes of index 2.

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 $\mathbb {F}_{q^{2m}}^{n}$ .

Combinatorial designs are closely related to linear codes. Recently, some near MDS codes were employed to construct <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-designs by Ding and Tang, which settles the question as to whether there exists an infinite family of near MDS codes holding an infinite family of <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-designs for <inline-formula> <tex-math notation="LaTeX">$t \geq 2$ </tex-math></inline-formula>. This paper is devoted to the construction of infinite families of 3-designs and 2-designs from special equations over finite fields. First, we present an infinite family of almost MDS codes over <inline-formula> <tex-math notation="LaTeX">${\mathrm{ GF}}(p^{m})$ </tex-math></inline-formula> holding an infinite family of 3-designs. We then provide an infinite family of almost MDS codes over <inline-formula> <tex-math notation="LaTeX">${\mathrm{ GF}}(p^{m})$ </tex-math></inline-formula> holding an infinite family of 2-designs for any field <inline-formula> <tex-math notation="LaTeX">${\mathrm{ GF}}(q)$ </tex-math></inline-formula>. In particular, some of these almost MDS codes are near MDS. Second, we present an infinite family of near MDS codes over <inline-formula> <tex-math notation="LaTeX">${\mathrm{ GF}}(2^{m})$ </tex-math></inline-formula> holding an infinite family of 3-designs by considering the number of roots of a special linearized polynomial. Compared to previous constructions of 3-designs or 2-designs from linear codes, the parameters of some of our designs are new and flexible.

Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study binary linear complementary dual $[n,k]$ codes with the largest minimum weight among all binary linear complementary dual $[n,k]$ codes. We characterize binary linear complementary dual codes with the largest minimum weight for small dimensions. A complete classification of binary linear complementary dual $[n,k]$ codes with the largest minimum weight is also given for $1 \le k \le n \le 16$.

Theory of additive complementary dual codes, constructions and computations

Remark on subcodes of linear complementary dual codes