Galois self-orthogonal constacyclic codes over finite fields

作者：Fu, Yuqing; Liu, Hongwei^*

来源：Designs, Codes, and Cryptography, 2022, 90(11): 2703-2733.

DOI：10.1007/s10623-021-00957-6

• Constacyclic code
• Galois self-orthogonal code
• Galois self-dual code
• Enumeration
• Weight distribution
• MDS code

Let F-q be a finite field with q = p(e) elements, where p is a prime and e is a positive integer. In 2017, Fan and Zhang introduced l-Galois inner products on the n-dimensional vector space F-q(n) for 0 <= l < e, which generalized the Euclidean inner product and Hermitian inner product. l-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes and Hermitian self-orthogonal codes, and can be used to construct entanglement-assisted quantum error-correcting codes. In this paper, we study l-Galois self-orthogonal constacyclic codes of length n over the finite field F-q. Sufficient and necessary conditions for constacyclic codes of length n over F-q being l-Galois self-orthogonal and l-Galois self-dual are characterized. A sufficient and necessary condition for the existence of nonzero l-Galois self-orthogonal constacyclic codes of length n over F-q is obtained. Formulae to enumerate the number of l-Galois self-orthogonal and l-Galois self-dual constacyclic codes of length n over F-q are found. In particular, formulae to enumerate the number of Hermitian self-orthogonal and Hermitian self-dual constacyclic codes of length n over F-q are obtained. Weight distributions of two classes of l-Galois self-orthogonal constacyclic codes are calculated. A family of MDS l-Galois self-orthogonal constacyclic codes over F-q is constructed.