Characterization and classification of optimal LCD codes

Abstract

Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. We give a characterization of LCD codes over ${\mathbb{F}}_q$ having large minimum weights for $q\in \{2,3\}$. Using the characterization, we determine the largest minimum weights among LCD $[n,k]$ codes over ${\mathbb{F}}_q$ for $(q,k)\in \{(2,4),(3,2),(3,3)\}$. Moreover, we give a complete classification of optimal LCD $[n,k]$ codes over ${\mathbb{F}}_q$ for $(q,k)\in \{(2,3),(2,4),(3,2),(3,3)\}$.

33 pages