A New Class of Linear Correcting Codes

A class of binary error-correcting codes is described. Each code in the class is specified by some polynomial in $GF(2^m)$. If the degree $t$ of the polynomial is known, the following estimate can be obtained for the code parameters: $n\leq 2^m$, $k\geq n-mt$, $d\geq 2t+1$. The codes described are in general noncyclic. The only cyclic codes in the class in question is the Bose–Chaudhuri–Hoquingham (BCH) code. All the basic properties of the BCH code are evidently the result of the fact that it belongs to this class of codes and not to the class of cyclic codes. For all the codes of the class in question, therefore, there exists a decoding scheme analogous to Peterson's algorithm for BCH codes. The codes are constructed by identifying the initial space of binary vectors with some set of rational functions.