On Dual Relationships of Secrecy Codes

We investigate properties of finite blocklength codes and their duals when used for coset coding over the binary erasure wiretap channel (BEWC). We identify sufficient conditions, related to the ranks of sub-matrices of a generator matrix that codes may satisfy to achieve the maximum equivocation among all codes with given blocklength and dimension, irrespective of the eavesdropper's channel erasure probability. We point out that binary maximum distance separable (MDS) codes are optimal for secrecy and we also show that simplex codes (and Hamming codes) have higher equivocation than families of codes with a single repeated column in the generator matrix (parity-check matrix). We conjecture that simplex and Hamming codes are optimal when used as the base linear code in a coset coding scheme for secrecy over the BEWC.

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