Algebraic Decoding of Reed-Solomon and BCH Codes

where α ∈ Fqm is an element of order n and gcd(b, n) = 1. This is a cyclic Reed-Solomon (RS) code with dmin = n − k + 1. Adding roots at all the conjugates of { α, α, . . . , αa+(n−k)b } (w.r.t. the subfield K = Fq) also allows one to define a length-n BCH subcode over Fq with dmin ≥ n−k+ 1. Also, any decoder that corrects all patterns of up to t = b(n− k)/2c errors for the original RS code can be used to correct the same set of errors for any subcode. The RS code operates by encoding a message polynomial m(x) = ∑k−1 j=0 mix i of degree at most k−1 into a codeword polynomial c(x) = ∑n−1 j=0 cix i using