Codes and decoding on graphs

We look at graphical descriptions of block codes known as trellises, which illustrate connections between algebra and graph theory, and can be used to develop powerful decoding algorithms. Trellises for linear block codes are known to grow exponentially with the code parameters and hence decoding on such objects is often infeasible. Of considerable interest to coding theorists therefore, are more compact descriptions called tail-biting trellises which in some cases can be much smaller than any ordinary trellis for the same code. We derive some interesting properties of tail-biting trellises and present an optimal maximum-likelihood decoding algorithm that performs rather well in terms of decoding complexity even at low signal-to-noise ratios