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Extra resources for Coding Theory and Cryptography
In this case Î²3 is a primitive 5th root of unity, and Î²5 is a primitive 3rd root of unity. We wish to construct cyclic Reed-Solomon codes, but first we need to establish that two polynomials will represent the same function on S if and only if they are equivalent (mod 1 + xn). 7 Let p(x), q(x) â GF(2r)[x], and S â GF(2r) be the set of nth roots of unity. , p(Î²i) = q(Î²i), for all Î²i â S) if and only if p(x) â ¡ q(x) (mod 1 + xn). Proof: Let q(x) = h(x)(1 + xn) + p(x), where degree (p(x)) < n.
We will use IMLD for the examples and exercises in this section, and throughout most of the rest of the text. We emphasize that MLD does not always work; in particular, if too many errors were made in transmitting across the BSC, then MLD fails. 3, has the greatest associated probability Ï p(v,w) among all codewords, so is the most likely codeword sent. 4 demonstrates this. ,Î±t be non-zero elements of GF(2r). Then 34 35 Proof: If Î±i = Î±j for some i â j then two rows of the matrix are identical, so the determinant is zero.
C) s0 = 1,s1 = 1,s2 = 1,s3 = 1,s4 = 1,s5 = 1,s6 = 1, and s7 = 1. (d) s0 = Î²10,s1 = Î²3,s2 = Î²13,s3 = Î²3,s4 = Î²12,s5 = Î²5,s6 = Î²13, and s7 = Î²3. (e) s0 = Î²12,s1 = Î²8,s2 = 0,s3 = Î²7,s4 = Î²13,s5 = Î²4,s6 = Î²13, and s7 = 1. (f) s0 = Î²2,s1 = 0,s2 = 0,s3 = Î²2,s4 =0,s5 =0,s6 =Î²2, and s7 = 0. 5 (proof of Berlekamp-Massey) (a) Prove recursively that deg (Pi(x)) â ¤iâ Di/2 (nothing that Pi(0) = 1). (b) Prove recursively that deg (Ri(x)) â ¤iâ â (1 + Di)/2 , after first showing that the choice of qi(0) forces the coefficient of xi in Pi(x)q-1(x) to be zero.