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 $6 \le k \le n-6$.

翻译：线性互补对偶（LCD）码可为双用户二进制加法信道提供最优线性编码方案。LCD码亦可用于抵御侧信道攻击与故障非侵入式攻击。令$d_{LCD}(n, k)$表示二元$[n,k, d]$ LCD码存在时$d$的最大值。在文献\cite{BS21}中，Bouyuklieva猜想对于任意长度$n$和维数$k \ge 2$，均有$d_{LCD}(n+1, k)=d_{LCD}(n, k)$或$d_{LCD}(n, k) + 1$。本文首先通过从二元$[n+1,k,d]$ $LCD_{o,e}$码构造二元$[n,k,d-1]$ LCD码（当$d \ge 3$且$k \ge 2$时），证明了Bouyuklieva的猜想\cite{BS21}。随后通过扩展码给出二元LCD码的距离下界，并利用该界及删减、缩短、扩展与延拓等方法构造了若干新型二元LCD码。最后，我们改进了长度$38 \le n \le 40$、维数$9 \le k \le 15$范围内部分已知的$d_{LCD}(n, k)$值，同时获得了$41 \le n \le 50$且$6 \le k \le n-6$范围内的若干$d_{LCD}(n, k)$值。