New Euclidean and Hermitian Self-Dual Cyclic Codes with Square-Root-Like Minimum Distances

TL;DR

This paper constructs new families of Euclidean and Hermitian self-dual cyclic codes over finite fields with asymptotically large minimum distances, advancing the theory of optimal error-correcting codes.

Contribution

It introduces novel constructions of Euclidean and Hermitian self-dual cyclic codes with square-root-like minimum distances over various finite fields.

Findings

Constructed Euclidean self-dual codes with minimum distance ≥ √(2^{s-1}n)-2^s.

Constructed Hermitian self-dual codes with minimum distance ≥ √(n/2).

Codes have large automorphism groups and asymptotically optimal minimum distances.

Abstract

Binary self-dual codes with large minimum distances, such as the extended Hamming code and the Golay code, are fascinating objects in the coding theory. They are closely related to sporadic simple groups, lattices and invariant theory. A family of binary self-dual repeated-root cyclic codes with lengths $n_i$ and minimum distances $d_i \geq \frac{1}{2} \sqrt{n_i+2}$, $n_i$ goes to the infinity for $i=1,2, \ldots$, was constructed in a paper of IEEE Trans. Inf. Theory, 2009. In this paper, we construct families of Euclidean self-dual repeated-root cyclic codes over the field ${\bf F}_{2^s}$, $s \geq 2$, with lengths $n_i$ and minimum distances at least $\sqrt{2^{s-1}n}-2^s$, where lengths $n_i$ go to the infinity. We also construct families of Hermitian self-dual repeated-root cyclic codes over the field ${\bf F}_{2^{2s}}$, $s \geq 1$, with lengths $n_i$ and minimum distances at least $\sqrt{n_i/2}$, where lengths $n_i$ go to the infinity. Our results show that Euclidean and Hermitian self-dual codes with large automorphism groups and large minimum distances can always be constructed.