Polynomial Codes Over Certain Finite Fields

A programmable logic device (130) as may be used in a communication System device Such as a digital Subscriber line modem (408) to perform Reed-Solomon decoding upon a received frame of digital values is disclosed. The program mable logic device (130) may be implemented as a DSP (130) or a general purpose microprocessor, for example. According to one disclosed embodiment of the invention, a group of look-up tables (60) are arranged, each look-up table (60) associated with one of the possible power values of a finite field, number up to twice the number of correctable errors. The contents of each entry (SYN) of the look-up tables (60) correspond to the finite field (e.g., Galois field) multiplication of a primitive element raised to an index power with a character of the finite field alphabet. Galois field multiplications (62) in Syndrome accumulation may now be performed with a single table look-up operation. According to other disclosed embodiments of the invention, look-up tables (60,160) are similarly arranged to contain the contents of finite field (e.g., Galois field) multiplication products for use in a Chien Search procedure. In a single thread version of the disclosed Chien Search procedure, a Subset of the same look-up tables (60) as used in syndrome accumulation may be utilized.