A family of self-orthogonal divisible codes with locality 2

TL;DR

This paper constructs a new family of self-orthogonal divisible linear codes with locality 2 using finite field trace and norm functions, providing optimal and near-optimal codes for distributed storage and lattice construction.

Contribution

Introduces a novel family of self-orthogonal divisible codes with specific weight distributions and locality 2, expanding coding options for storage and cryptography.

Findings

Codes have 3, 4, or 5 nonzero weights.

Existence of infinite distance-optimal binary codes.

Codes applicable in distributed storage and lattice construction.

Abstract

Linear codes are widely studied due to their applications in communication, cryptography, quantum codes, distributed storage and many other fields. In this paper, we use the trace and norm functions over finite fields to construct a family of linear codes. The weight distributions of the codes are determined in three cases via Gaussian sums. The codes are shown to be self-orthogonal divisible codes with only three, four or five nonzero weights in these cases. In particular, we prove that this family of linear codes has locality 2. Several optimal or almost optimal linear codes and locally recoverable codes are derived. In particular, an infinite family of distance-optimal binary linear codes with respect to the sphere-packing bound is obtained. The self-orthogonal codes derived in this paper can be used to construct lattices and have nice application in distributed storage.