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IEEE Transactions on Information Theory

Several Families of Self-Orthogonal Codes and Their Applications in Optimal Quantum Codes and LCD Codes

Xinran Wang, Ziling Heng

Published Nov 13, 2023• 16 citations
Publisher

Abstract

Self-orthogonal codes have nice applications in many areas including quantum codes, lattices and LCD codes. For a prime powerq, it is in general difficult to constructq-ary self-orthogonal codes. In the literature, there exists no simple method to judge whether a generalq-ary linear code is self-orthogonal or not. In this paper, we mainly present several families ofq-ary self-orthogonal codes and study their applications in quantum codes and LCD codes. Firstly, several families ofq-ary linear codes are constructed by some special defining sets. These codes are proved to be self-orthogonal. To this end, we determine the numbers of solutions of some systems of equations over finite fields. Secondly, three families ofq-ary quantum codes with unbounded length and minimum distance three are constructed from the self-orthogonal codes. These quantum codes are optimal according to the quantum Hamming bound. In particular, some of them have better parameters than known ones. Thirdly, several families ofq-ary LCD codes are constructed from the self-orthogonal codes. Many optimal or almost optimal binary and ternary LCD codes are produced by our constructions. Some binary and ternary LCD codes have better parameters than known ones.

DOI
10.1109/tit.2023.3332332

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Self-orthogonal codes have nice applications in many areas including quantum codes, lattices and LCD codes. For a prime powerq, it is in general difficult to constructq-ary self-orthogonal codes. In the literature, there exists no simple method to judge whether a generalq-ary linear code is self-orthogonal or not. In this paper, we mainly present several families ofq-ary self-orthogonal codes and study their applications in quantum codes and LCD codes. Firstly, several families ofq-ary linear codes are constructed by some special defining sets. These codes are proved to be self-orthogonal. To this end, we determine the numbers of solutions of some systems of equations over finite fields. Secondly, three families ofq-ary quantum codes with unbounded length and minimum distance three are constructed from the self-orthogonal codes. These quantum codes are optimal according to the quantum Hamming bound. In particular, some of them have better parameters than known ones. Thirdly, several families ofq-ary LCD codes are constructed from the self-orthogonal codes. Many optimal or almost optimal binary and ternary LCD codes are produced by our constructions. Some binary and ternary LCD codes have better parameters than known ones.
It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.
Abstract Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal q2q^2q2 q 2 -ary linear codes. This result can be generalized to q2mq^{2 m}q2m q 2 m -ary linear codes, m>1m > 1m>1 m > 1 . We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de , 2020) and new q-ary ones, with q≠2q \ne 2q=2 q ≠ 2 , improving others in the literature.
Self-orthogonal codes have attracted considerable attention owing to their applications in quantum error-correcting codes, linear complementary dual codes, and a variety of other fields. In this paper, we construct new families of self-orthogonal codes and self-orthogonal minimal codes by establishing criteria that characterize the self-orthogonality of certain linear codes. To this end, we first establish several new criteria for linear codes arising from the defining-set construction to be self-orthogonal. More specifically, for q>3q > 3q>3, we show that the code CD\mathcal{C}_DCD​ is self-orthogonal whenever the defining set DDD is GGG-invariant, where G⊆Fq∗G \subseteq \mathbb{F}_q^*G⊆Fq∗​ and ∣G∣>2|G| > 2∣G∣>2. For q=2,3q = 2, 3q=2,3, we characterize the self-orthogonality of CD\mathcal{C}_DCD​ using certain character sums. By combining these criteria with partial difference sets, we construct several new families of self-orthogonal codes, which lead to optimal or almost optimal quantum codes. Secondly, via the action of a multiplicative subgroup G⊆Fq∗G \subseteq \mathbb{F}_q^*G⊆Fq∗​ with ∣G∣>2|G| > 2∣G∣>2 on the columns of a projective linear code, we construct self-orthogonal codes with flexible parameters. From their augmented codes, we derive quantum codes. By choosing suitable projective codes, we obtain optimal quantum codes with high parametric flexibility. Thirdly, we employ the characteristic function method to construct linear codes and establish criterion for their self-orthogonality. Based on this, we construct several classes of self-orthogonal minimal codes that violate the Ashikhmin-Barg condition, using vectorial dual-bent functions and sss-plateaued functions.