On the Existence of Optimal Binary LCD Codes Under Hierarchical Poset Metrics

Abstract

A linear code is referred to as a linear complementary dual (LCD) code when it has only a trivial intersection with its dual. LCD codes have gained prominence in research due to their application in cryptography, communication systems, and data storage.  ”This article explores the binary LCD hierarchical poset code, wherein the dimension is determined by the rank of the Gramian of its generator matrix. By employing the canonical systematic form of the generator matrix of the hierarchical poset code, the corresponding Gramian matrix is specified by imposing certain conditions on the support of basis elements. Utilizing the Griesmer bound of linear code under Hamming metric and the canonical decomposition of the hierarchical poset code, a upper bound is established on the maximum distance of the hierarchical poset code with any hull dimension under specified conditions. Furthermore, the study investigates the existence of optimal binary LCD codes under a hierarchical poset metric when $n$ is equivalent to $a$ mod $8$ where, $a$ ranges from $0$ to $7$.”