Constructing valid convex hull inequalities for single parity-check codes over prime fields
In this work, we consider adaptive linear programming (LP) decoding of ternary linear codes, i. e., linear codes over the finite field F<sub>q</sub> with q = 3 elements. In particular, we characterize completely the codeword polytope (or the convex hull) of the binary image, under Flanagan's embedding, of a ternary single parity-check code. Then, this characterization is used to develop an efficient adaptive LP decoder for ternary codes. Numerical experiments confirm that this decoder is very efficient compared to a static LP decoder and scales well with both block length and check node degree. Finally, we briefly consider the case of nonbinary codes over the finite field F<sub>q</sub> with q = 3<sup>m</sup> elements, where m > 1 is a positive integer.