New distance bounds for quasi-cyclic codes

Eric Zhi Chen

P. Heijnen, Henk van Tilborg, S. Weijs

Many binary quasi-cyclic codes have a better minimum distance than other known linear codes with the same parameters. We present 43 new quasi-cyclic codes, which improve on the parameters of previously found quasi-cyclic codes. Five of them even improve the lower bound on the minimum distance for linear codes. Furthermore we present a decoding algorithm for the binary quasi-cyclic codes of rate 1/m.

P. Heijnen, Henk C. A. van Tilborg

Kanat Abdukhalikov, Tushar Bag, Daniel Panario

Zhi Chen

Twelve new binary quasi-cyclic codes, which improve the lower bounds on minimum distances for binary linear codes, are presented, and a Web database on best-known binary quasi-cyclic codes is constructed for public access.

Yiting Liu, Chaofeng Guan, Chao Du +1 more

This paper considers three kinds of quasi-cyclic codes of index two with one generator or two generators and their applications in quantum code construction. In accordance with the algebraic structure of linear codes, we determine the lower bounds of the symplectic weights of these quasi-cyclic codes. Quasi-cyclic codes with the dual-containing property enable the construction of quantum codes. Defining the coefficient symmetric polynomials of the generator polynomials gives a concise condition for the dual-containing of the quasi-cyclic codes. The lower bound results can significantly reduce the scope of the search for a larger minimum distance of quasi-cyclic codes. With these algebraic results and computer supports, we obtain classical quasi-cyclic codes with better parameters and some new quantum codes under the symplectic construction. In particular, two examples of the new quantum codes [[63,42,6]]2,[[51,35,5]]2 improve the corresponding codes in Grassl’s code table.