The hull of two classical propagation rules and their applications

Propagation rules are of great help in constructing good linear codes. Both Euclidean and Hermitian hulls of linear codes perform an important part in coding theory. In this paper, we consider these two aspects together and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the direct sum construction and the $(\mathbf{u},\mathbf{u+v})$-construction. Some new criteria for resulting codes derived from these two propagation rules being self-dual, self-orthogonal or linear complement dual (LCD) codes are given. As applications, we construct some linear codes with prescribed hull dimensions and many new binary, ternary Euclidean formally self-dual (FSD) LCD codes, quaternary Hermitian FSD LCD codes and good quaternary Hermitian LCD codes which are optimal or have best or almost best known parameters according to Datebase at $http://www.codetables.de$. Moreover, our methods contributes positively to improve the lower bounds on the minimum distance of known LCD codes.Comment: 16 pages, 5 table

On MDS Codes With Galois Hulls of Arbitrary Dimensions

The Galois hulls of linear codes are a generalization of the Euclidean and Hermitian hulls of linear codes. In this paper, we study the Galois hulls of (extended) GRS codes and present several new constructions of MDS codes with Galois hulls of arbitrary dimensions via (extended) GRS codes. Two general methods of constructing MDS codes with Galois hulls of arbitrary dimensions by Hermitian or general Galois self-orthogonal (extended) GRS codes are given. Using these methods, some MDS codes with larger dimensions and Galois hulls of arbitrary dimensions can be obtained and relatively strict conditions can also lead to many new classes of MDS codes with Galois hulls of arbitrary dimensions.Comment: 21 pages,5 table

On Galois self-orthogonal algebraic geometry codes

Galois self-orthogonal (SO) codes are generalizations of Euclidean and Hermitian SO codes. Algebraic geometry (AG) codes are the first known class of linear codes exceeding the Gilbert-Varshamov bound. Both of them have attracted much attention for their rich algebraic structures and wide applications in these years. In this paper, we consider them together and study Galois SO AG codes. A criterion for an AG code being Galois SO is presented. Based on this criterion, we construct several new classes of maximum distance separable (MDS) Galois SO AG codes from projective lines and several new classes of Galois SO AG codes from projective elliptic curves, hyper-elliptic curves and hermitian curves. In addition, we give an embedding method that allows us to obtain more MDS Galois SO codes from known MDS Galois SO AG codes.Comment: 18paper

Several classes of Galois self-orthogonal MDS codes

Let $q=p^h$ be an odd prime power and $e$ be an integer with $0\leq e\leq h-1$. $e$-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes ($e=0$) and Hermitian self-orthogonal codes ($e=\frac{h}{2}$ and $h$ is even). In this paper, we propose two general methods of constructing several classes of $e$-Galois self-orthogonal generalized Reed-Solomn codes and extended generalized Reed-Solomn codes with $2e\mid h$. We can determine all possible $e$-Galois self-orthogonal maximum distance separable codes of certain lengths for each even $h$ and odd prime number $p$.Comment: 18 pages, 9 table