On Binary Cyclic Codes Which are Also Cyclic Codes Over $GF(2^s )$

Abstract

Introduction. Most encoding and decoding equipment operates in binary symbols, whereas it is often desirable to use a code consisting of symbols from GF(2S); each code symbol is really s binary symbols. This happens, for example, in multilevel transmission; it also happens when the chief function of the code is burst-error correction. The question discussed in this paper is when and how a binary cyclic code of block length s(2s 1) can be mapped in a way onto a cyclic code of block length 2s 1 over GF(2S). Such codes are attractive because they combine the advantages of multilevel efficiency with binary implementation; however the moral of this paper seems to be that we had better look for other ways to achieve this goal. Indeed the natural practical mapping between codes over GF(2) and codes over GF(2S) is mathematically rather unnatural. The image of a cyclic code over GF(2) may easily fail to be a linear space over GF(2S). Thus it is not surprising (although disappointing) that we appear to find only a rather small class of rather poor codes. The only previous published work on this subject known to the writer is due to M. Hanan and F. P. Palermo; these authors discuss a more general form of the same basic question, and arrive at the same basic conclusion. The plan of this paper is as follows: Section 1 describes the mapping p from the vector space over the binary field to the vector space over GF(2s), and gives an explicit form for the inverse mapping. Section 2 gives necessary and sufficient conditions for the image under (p of a binary cyclic code to be a cyclic code over GF(2S), and consequently for the image under p ` of a cyclic code over GF(2S) to be a binary cyclic code. These conditions are essentially the same as those given by Hanan and Palermo, and are definitely unhelpful from a practical point of view. Section 3 describes the class of codes, which can always be mapped back and forth, and shows that the union and intersection of an interlaced code with a mappable code is again mappable. Section 4 shows how to get new mappings from old-specifically, if a(x) is the polynomial of least degree in a mappable cyclic code (the generator polynomial), then the reciprocal polynomial of a(x) is also mappable, and so is the polynomial obtained by reversing the order of the coefficients of a(x). (The mapping function 5p is different in the three cases.) Section 5 describes the one choice of p which we know always works and shows how it works. Section 6 contains several theorems which are useful when looking for mappings; for example, we need consider only noninterlaced codes generated over GF(2s) by polynomials of degree ?s 1. The Appendix contains a series of examples, which hopefully will be useful as illustrations of the text.