New Constructions of Optimal Binary LCD Codes

TL;DR

This paper proves a conjecture about the maximum distance of binary LCD codes, introduces new construction methods, and improves known bounds for various code parameters.

Contribution

It proves Bouyuklieva's conjecture on the distance evolution of LCD codes and develops new construction techniques for optimal binary LCD codes.

Findings

Proved Bouyuklieva's conjecture for LCD codes.

Constructed new binary LCD codes with improved parameters.

Provided lower bounds for the minimum distance of LCD codes.

Abstract

Linear complementary dual (LCD) codes can provide an optimum linear coding solution for the two-user binary adder channel. LCD codes also can be used to against side-channel attacks and fault non-invasive attacks. Let $d_{LCD}(n, k)$ denote the maximum value of $d$ for which a binary $[n,k, d]$ LCD code exists. In \cite{BS21}, Bouyuklieva conjectured that $d_{LCD}(n+1, k)=d_{LCD}(n, k)$ or $d_{LCD}(n, k) + 1$ for any lenth $n$ and dimension $k \ge 2$. In this paper, we first prove Bouyuklieva's conjecture \cite{BS21} by constructing a binary $[n,k,d-1]$ LCD codes from a binary $[n+1,k,d]$ $LCD_{o,e}$ code, when $d \ge 3$ and $k \ge 2$. Then we provide a distance lower bound for binary LCD codes by expanded codes, and use this bound and some methods such as puncturing, shortening, expanding and extension, we construct some new binary LCD codes. Finally, we improve some previously known values of $d_{LCD}(n, k)$ of lengths $38 \le n \le 40$ and dimensions $9 \le k \le 15$. We also obtain some values of $d_{LCD}(n, k)$ with $41 \le n \le 50$ and $6 \le k \le n-6$.