Systematic byte-oriented codes

Abstract

The order (number of bits or length) of classical cycliccodes is usually not a multiple of an integer number ofbytes, which leads to unproductive expenditures of computingresources with their hardware-software implementation.For this reason, the transition to byte-orientedcodes, in which both the length k of information wordsI and the number r of test bits are multiples R of aninteger number of bytes, seems most appropriate for practicaluse. A distinctive feature of the proposed approach tosynthesis (information coding) and code analysis (messagedecoding) is the rejection of generators G and verificationmatrices H , usually accompanying systematic cycliccodes, and their replacement by a single matrix P of paritysymbols (MAP), smaller in volume compared to using matricesG and H . The basis for the formation of MPS cyclic(n, k, t)  codes, where n  the code length and t the multiplicity of eliminated errors in codewords, are generators(generating) polynomials (one-dimensional binaryvectors), denoted by the symbol b . A binary polynomial ofr  degree is a generating polynomial of a primitive cyclic(n, k, t)  code if and only if the so-called "control" (k 1)  string k 1 s  , which is an extension of the matrixof code P parity symbols and computed according to therules of forming the rows of this matrix, but not enteringinto it, is determined by the relation  11 0 1 rk s   (necessaryconditions ), and the weight  of each row of the paritymatrix is not less than 2t , and the Hamming i , j d distancebetween any pairs of rows ( , ) i j s s of the matrix P is suchthat , 2 1 i j d  t  (sufficient conditions). The operators ofinverse permutation of rows and columns of the matricesmutually relate dual matrixes of parity symbols, i.e. the matricesgenerated by the dual binary polynomials. A systematicnoise-proof (16, 8, 2)  code is generated, generated bya symmetric irreducible polynomial of the eighth degreeb 100111001, which is unique (that is, unique in its kind)and optimal in a class of byte-oriented codes for a numberof criteria. A detailed characteristic of the algorithm forsyndrome decoding of byte-oriented codes is given.