Cross Parity Check Convolutional Codes

Abstract

A class of convolutional codes called cross parity check (CPC) codes, useful for the protection of data stored on magnetic tape, is described and analyzed. CPC codes are first explained geometrically; their construction is described in terms of constraining data written onto a tape in such a way that, when lines of varying slope are drawn across the tape, the bits falling on those lines sum to zero modulo two. This geometric interpretation is then formalized by the construction of canonical parity check matrices and systematic generator matrices for CPC codes and by computing their constraint lengths. The distance properties of CPC codes are analyzed, and it is shown that these codes are maximum distance separable (MDS) convolutional codes. In addition, examples are given of both error and erasure decoding algorithms that take advantage of the geometric regularity of cross parity check codes. Finally, the technique of parity check matrix reduction-useful for reducing the inherent decoding delay of CPC codes-is described. This technique consists of dividing each term of the parity check matrix by some polynomial and retaining only the remainder. A class of polynomials which are particularly attractive for this purpose is identified. © 1989 IEEE