On a class of optimal nonbinary linear unequal-error-protection codes for two sets of messages

We intraduce the n-dimensional key equation, which exhibitsthe error-locator polynomial of an n-dimensional cyclic code as a productof n univariate polynomials and the error-evaluator polynomial as an n-variable polynomial. We then reinterpret these polynomials in the contextof linear recurring sequences. In particular, we reduce the decodingproblem to successive application of the BerlekampMassey algorithm.With this new method, we are able to decode (up to half their mini"distance) many codes in a table of 2-Dcyclic codes due to Jensen. I. INTRODUCITONAND OTATIONLet n 2 2, K be a finite field and K[X] = K [ X 1 , .. . ,X,]. An n-dimensional (n-D) cyclic or abelian code is an ideal in the polynomialalgebra K [ X ] / ( X p-l,.-.,X,N"1).See [1]-[3] for details. Weconsider the problem of decoding these codes. Our approach is basedon generalizing the key equation to n dimensions and successiveapplication of the ordinary BerlekampMassey algorithm. We giveseveral examples of our algorithm at work; all of the 2-D cyclic codesin Jensen's table [8], whose minimum distance does not exceed eight,can be decoded.In more detail, let