Some important results on 𝒯-Direct codes

Abstract

[Formula: see text]-Direct codes are an extension to the class of linear codes having complementary duals (LCD codes). Defined over a finite field [Formula: see text], it is comprised of [Formula: see text] linear codes [Formula: see text], [Formula: see text] with [Formula: see text], where [Formula: see text] is the dual with respect to [Formula: see text]. A 2-Direct code [Formula: see text] with respect to [Formula: see text] is comprised of only LCD codes: [Formula: see text]. On the contrary, two LCD codes do not set up a [Formula: see text]-Direct code in general. This paper presents some important and generalized results on [Formula: see text]-Direct codes, in that it attempts to construct [Formula: see text]-Direct codes from LCD codes. The class of [Formula: see text]-cyclic maximum rank distance (MRD) codes as having complementary duals over [Formula: see text] are generalized. Dual bases such as self-dual basis and self-dual normal basis play a crucial role in constructions. Further, construction implausibility of [Formula: see text]-Direct codes from almost self-dual bases is also dealt. Results obtained are demonstrated through examples.