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By Jonas Eriksson.

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Turning out attention back to the Sudan polynomials we will study them 36 Chapter 3. 3: Value of strain parameter s needed to achieve maximum attainable list decoding radius with Sudan’s algorithm, plotted versus the code dimension k. The displayed curves are for Reed-Solomon codes of length 25, 26, 31, 62, 63 and 255. Results obtained through computer search. 3. Sudan’s algorithm 37 more closely. Specifically we are interested in what properties a bivariate polynomial must have in order to be a Sudan polynomial.

His solution was later given an elegant formulation by Massey [61] who recognized BCH decoding as a problem in the design of linear-feedback shift registers. The algorithm is now referred to as the Berlekamp-Massey algorithm. An intermediate step σ (k) , ω (k) , ν (k) of the solution in the iterative algorithm can, for iteration step k, be written as a solution to the equation (1 + S(z))σ (k) (z) ≡ ω (k) (z) mod z k+1 . 30 Chapter 3. List decoding of Reed-Solomon codes The algorithm successively adjusts the solutions to accommodate for the 2t known syndromes S1 , S2 , .

For a specific code it counts the number of error patterns of weight i which the list decoder will not be able to correct. That is, those error patterns of weight i outside the set of correctable error patterns. We note that these error patterns cause a list decoder to produce an output list on which the sent codeword is not the uniquely closest codeword. (τ ) We will refer to the sequence βi , i = 0, 1, . . , n as the distribution of elusive error patterns. Since we know the total number of possible error patterns of a given weight, we also know the number of error patterns of a given weight (τ ) within the set of correctable error patterns given βi .

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