On the minimum distances of binary optimal LCD codes with dimension 5

Abstract

Let $ d_{a}(n, 5) $ and $ d_{l}(n, 5) $ be the minimum weights of optimal binary $ [n, 5] $ linear codes and linear complementary dual (LCD) codes, respectively. This article aims to investigate $ d_{l}(n, 5) $ of some families of binary $ [n, 5] $ LCD codes when $ n = 31s+t\geq 14 $ with $ s $ integer and $ t \in\; \{2, 8, 10, 12, 14, 16, 18\} $. By determining the defining vectors of optimal linear codes and discussing their reduced codes, we classify optimal linear codes and calculate their hull dimensions. Thus, the non-existence of these classes of binary $ [n, 5, d_{a}(n, 5)] $ LCD codes is verified, and we further derive that $ d_{l}(n, 5) = d_{a}(n, 5)-1 $ for $ t\neq 16 $ and $ d_{l}(n, 5) = 16s+6 = d_{a}(n, 5)-2 $ for $ t = 16 $. Combining them with known results on optimal LCD codes, $ d_{l}(n, 5) $ of all $ [n, 5] $ LCD codes are completely determined.

Keywords

• optimal code
• lcd code
• hull dimension
• defining vector
• reduced code