Rank and Kernel of a Binary Code

Given two binary codes of length n, we construct a code of length 2n. The construction works for linear and nonlinear codes. For the linear codes, we obtain that the dimension of the final code is 2k 1 + k 2 − dim(C 1 ∩ C 2), which dim(C 1) = k 1, and dim(C 2) = k 2. In this paper, we get that both parameters are the relation of the corresponding ones of the starting codes.

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[2]  Morris Plotkin,et al.  Binary codes with specified minimum distance , 1960, IRE Trans. Inf. Theory.