Maximum Likelihood Decoding of Codes on the Asymmetric Z-channel

AbstractThe aim of this paper is to extend some basic concepts related to the MaximumLikelihood decoding of codes on the Z-channel, which is a particular, but very important,example of an asymmetric channel. We study distance properties of linear codes overthe Z-channel, in order to de ne a suitable metric for the implementation of a MaximumLikelihood decoder on the channel. A combinatorial expression for an upper bound on theprobability of incorrect Maximum Likelihood decoding is also provided, and comparisonsare given to evaluate the tightness of the bound, when a Hamming code, a Turbo codeand an LDPC code are decoded on the asymmetric channel. 1 Introduction Errors induced by communications on a noisy channel or storage medium can be reduced to anydesired level by proper encoding of the information, as long as the information rate is less thanthe capacity of the channel. This fact was shown in a landmark paper by Shannon, in 1948 [1];since that time, much work has been devoted to the problem of devising ecient encoding anddecoding methods for error control in a noisy environment.In a block-coded system, the source output u represents a k-bit message, the encoder outputc represents an n-symbol codeword, the demodulator output r represents the correspondingbinary received n-tuple and the decoder output u^ represents the k-bit estimate of the encodedmessage. The decoder must produce an estimate u^ of the information sequence u based onthe received sequence r. Equivalently, since there is a one-to-one correspondence between theinformation sequence u and the codeword c, the decoder can produce an estimate ^c of thecodeword c. Obviously, u^ = u if and only if ^c = c. The strategy adopted for choosing an