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

Self-orthogonal codes have nice applications in many areas including quantum codes, lattices and LCD codes. For a prime power
$q$
, it is in general difficult to construct
$q$
-ary self-orthogonal codes. In the literature, there exists no simple method to judge whether a general
$q$
-ary linear code is self-orthogonal or not. In this paper, we mainly present several families of
$q$
-ary self-orthogonal codes and study their applications in quantum codes and LCD codes. Firstly, several families of
$q$
-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
$q$
-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
$q$
-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.