T Direct Gf N

A T -Direct code is defined as the set of T F -ary linear codes Γ1,Γ2, . . . ,ΓT such that Γi ∩ Γi = {0}, where Γi = Γ1 ⊕ Γ2⊕ · · · ⊕ Γi−1⊕ Γi+1 ⊕ · · · ⊕ ΓT is the dual of Γi with respect to the direct sum Λ = Γ1 ⊕ Γ2⊕ · · · ⊕ ΓT for each i = 1, 2, . . . , T . In this paper, a construction to a class of T -Direct codes with the constituent codes from the class of 2-Cyclic Maximum Rank Distance codes is given. These class of codes are defined with a Rank metric. An application of the constructed class of codes over the T -user F -Adder Channel, namely a coding scheme for the noisy case, is given. In an attempt to show the usefulness of T -Direct codes as multi-user codes, a distance construction method for the class of T -Direct codes constituted by a set of T maximum rank distance codes is proposed. The proposed distance construction is shown to increase the minimum distance of the constituent codes of T -Direct codes when employed. Further, a construction procedure that constructs (n + m)-Direct codes over GF (2) is developed from the proposed distance construction, 0 < m ≤ n(n+1) 2 . It is shown that the extended distance construction procedure increases the number of users (constituent codes) of a T -Direct code; when employed on an n-Direct code, it constructs an [ n(n+1) 2 ] -Direct code, thereby supporting more users in a multi-user environment.