Linear complementary dual (LCD) codes are linear codes which intersect their dual codes trivially, which have been of interest and extensively studied due to their practical applications in computational complexity and information protection. In this paper, we give some methods for constructing LCD codes over small finite fields by modifying some typical methods for constructing linear codes. We show that all odd-like binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes can be constructed using the modified methods. Our results improve the known lower bounds on the largest minimum distances of LCD codes. Furthermore, we give two counterexamples to disprove the conjecture proposed by Bouyuklieva (Des. Codes Cryptogr. 89(11): 2445-2461, 2021).

翻译：线性互补对偶（LCD）码是一类与其对偶码平凡相交的线性码，因其在计算复杂性和信息保护中的实际应用而备受关注并得到广泛研究。本文通过改进若干构造线性码的典型方法，给出了在小有限域上构造LCD码的一些方法。我们证明，所有奇似二进制LCD码、三进制LCD码及四进制Hermitian LCD码均可通过这些改进方法构造。我们的结果改进了LCD码最大最小距离的已知下界。此外，我们给出了两个反例，以反驳Bouyuklieva提出的猜想（Des. Codes Cryptogr. 89(11): 2445-2461, 2021）。