New Constructions of Optimal Binary LCD Codes
Guodong Wang, Shengwei Liu, Hongwei Liu
TL;DR
The paper advances the theory and practice of binary LCD codes by proving Bouyuklieva’s conjecture through a construction that derives a $[n,k,d-1]$ LCD code from a $[n+1,k,d]$ $ ext{LCD}_{o,e}$ code (for $d\\ge 3$, $k\\ge 2$), and by establishing a distance bound via expanded codes. It develops two main construction techniques—expansion over field extensions and extension/puncturing-based methods—that preserve the LCD property and Hull structure, enabling systematic generation of longer and higher-distance LCD codes. Using these approaches, the authors produce new binary LCD codes and tighten known bounds for lengths $38\\le n\\le 50$ across several $k$ values, aided by extensive computer-aided searches. The work provides a practical toolkit for constructing optimal or near-optimal binary LCD codes with potential applications in communications and side-channel resistance, supported by concrete code tables and algorithmic strategies.
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 $6 \le k \le n-6$.