Optimal few-weight codes from simplicial complexes

TL;DR

This paper introduces new code constructions over a ring using simplicial complexes, resulting in binary codes with few weights that are optimal or meet bounds, expanding the design of efficient error-correcting codes.

Contribution

The paper presents two novel constructions of codes over f 2 + uf 2 using simplicial complexes, including the determination of Lee weight distributions and the creation of optimal binary codes.

Findings

Codes have few Lee weights.

Binary codes meet the Griesmer bound.

Some codes are distance optimal.

Abstract

Recently, some infinite families of binary minimal and optimal linear codes are constructed from simplicial complexes by Hyun {\em et al}. Inspired by their work, we present two new constructions of codes over the ring $\Bbb F_2+u\Bbb F_2$ by employing simplicial complexes. When the simplicial complexes are all generated by a maximal element, we determine the Lee weight distributions of two classes of the codes over $\Bbb F_2+u\Bbb F_2$. Our results show that the codes have few Lee weights. Via the Gray map, we obtain an infinite family of binary codes meeting the Griesmer bound and a class of binary distance optimal codes.