On Some Families of Codes Related to the Even Linear Codes Meeting the Grey–Rankin Bound

Abstract

Bounds for the parameters of codes are very important in coding theory. The Grey–Rankin bound refers to the cardinality of a self-complementary binary code. Codes meeting this bound are associated with families of two-weight codes and other combinatorial structures. We study the relations among six infinite families of binary linear codes with two and three nonzero weights that are closely connected to the self-complementary linear codes meeting the Grey–Rankin bound. We give a construction method and partial classification results for such codes. The properties of the codes in the studied families and their relations help us in constructing codes of a higher dimension from codes with a given dimension.

Some Results on Self-Complementary Linear Codes

“…. 0) T as the last column to all matrices from (1). In this way, we obtain codes with minimum distance 1, and from the set R, we obtain the corresponding set of codes R ′ with dimension k and minimum distance 1.…”

“…In our previous work [1], we studied binary linear self-complementary codes meeting the Grey-Rankin bound. We considered the relations among six infinite families of binary linear codes with two and three nonzero weights that are closely connected to these self-complementary codes.…”

“…For example, all self-dual codes are also self-complementary. Furthermore, the special classes of self-complementary codes that meet the Gray-Rankin bound and the two-and three-weight codes connected to them are of great interest [1,3,4]. Such codes are connected to many combinatorial structures such as SDP designs (a combinatorial design has the symmetric difference property, or is an SDP design if the symmetric difference of any two blocks is either a block or the complement of a block), bent functions, strongly regular graphs, and others.…”

“…There is a unique [16,5,8] self-complementary code, and this is the Reed-Muller code RM (1,4). The self-complementary [16,4,8] and [16,3,8] codes are subcodes of RM(1, 4).…”