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

Abstract

Self-orthogonal codes have nice applications in many areas including quantum codes, lattices and LCD codes. For a prime power <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> , it is in general difficult to construct <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary self-orthogonal codes. In the literature, there exists no simple method to judge whether a general <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary linear code is self-orthogonal or not. In this paper, we mainly present several families of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary self-orthogonal codes and study their applications in quantum codes and LCD codes. Firstly, several families of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -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 of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -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 of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -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.