Abstract
This paper describes a class of two-dimensional codes called cascaded Reed-Solomon (CRS) codes and an algorithm for decoding these codes up to their minimum distance. CRS codes are cascade (or generalized concatenated) codes in which Reed-Solomon codes are used for both the inner and outer codes. We introduce hyperbolic cascaded Reed-Solomon (HCRS) codes, which have maximal rate among CRS codes of a given minimum distance. Our algorithm decodes any CRS code to its minimum distance by calculating a Gröbner basis for an ideal which identifies the error locations. This error location algorithm is based on Sakata's algorithm, but with two significant modifications. First of all, the iterations and terms of polynomials are ordered according to the lexicographic ordering. Secondly, unknown syndromes are calculated as needed, by a simple threshold rule. Once the error locations are known, the error values can be calculated by solving an analog of the key equation for Reed-Solomon codes.
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