New Dual Relationships for Error-Correcting Wiretap Codes

In this paper, we consider the equivocation of finite blocklength coset codes when used over binary erasure wiretap channels. We make use of the equivocation matrix in comparing codes that are suitable for scenarios with noisy channels for both the intended receiver and an eavesdropper. Equivocation matrices have been studied in the past only for the binary erasure wiretap channel model with a noiseless channel for the intended recipient. In that case, an exact relationship between the elements of equivocation matrices for a code and its dual code was identified. The majority of work on coset codes for wiretap channels only addresses the noise-free main channel case, and extensions to noisy main channels require multi-edge type codes. In this paper, we supply a more insightful proof for the noiseless main channel case, and identify a new dual relationship that applies when two-edge type coset codes are used for the noisy main channel case. The end result is that the elements of the equivocation matrix for a dual code are known precisely from the equivocation matrix of the original code according to fixed reordering patterns. Such relationships allow one to study the equivocation of codes and their duals in tandem, which simplifies the search for best and/or good finite blocklength codes of various sizes. This paper is the first work that succinctly links the equivocation/error correction capabilities of dual codes for two-edge type coset coding over erasure-prone main channels.