Error-trellises for convolutional codes .I. Construction

An error-trellis is a directed graph that represents all the sequences belonging to the coset which contains the symbol-by-symbol detected version of a given received sequence. A modular construction of error-trellises for an (n,k) convolutional code over GF(q) is presented. The trellis is designed on the basis of partitioning the scalar check matrix of the code into submatrices of l rows, accompanied with a corresponding segmentation of the syndrome. The value of the design parameter l is an essentially unconstrained multiple of n-k. For all the cosets of the code, the sections of the error-trellis are drawn from a collection of only q^l modules; the module for each section is determined by the value of the associated syndrome segment. In case the construction is based on a basic polynomial check matrix, either canonical or noncanonical, then the error-trellis is minimal in the sense that σ⩽μ, where σ is the dimension of the state space of the trellis and μ is the constraint length of a canonical generator matrix for the code. For basic check matrices with delay-free columns, the inequality reduces to σ=μ