On a signal processing algorithms based class of linear codes

Abstract

Abstract. In this work, the correspondence between linear (n,k,d) codes and aperiodic convolution algorithms for computing a system 05 of k bilinear forms over GF(p") is explored. A number of properties are established for the linear codes that can be obtained from a computational procedure of this type. A particular bilinear form is considered and a class of linear codes over GF(2 m) is derived with varying k and d parameters. The code length n is equal to the multiplicative complexity of the computation of an aperiodic convolution and an efficient computation thereof leads to the shortest codes possible using this approach, many of which are optimal or near-optimal. A new decoding procedure for this class of linear codes is presented which exploits the block structure of the generator matrix of the codes. Several interesting observations are made on the nature of the codes obtained as a result of such computations. Such a computation of bilinear forms can be generalized to include other bilinear forms and the related classes of codes.