Coding and Cryptography by Tom W. Körner

By Tom W. Körner

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In particular, C will have length n and defining set A = {α, α2, . . , αδ−1 } where α is a primitive nth root of unity in K. Let t be the largest integer with 2t + 1 ≤ δ. We show how we can correct up to t errors. Suppose that a codeword c = (c0 , c1 , . . , cn−1 ) is transmitted and that the string received is r. We write e = r − c and assume that E = {0 ≤ j ≤ n − 1 : ej = 0} has no more than t members. In other words, e is the error vector and we assume that there are no more than t errors.

Then we can find a non-zero P0 ∈ P which divides every P ∈ P. Proof. Consider a non-zero polynomial P0 of smallest degree in P. Recall that the polynomial P (X) = X 2 + 1 has no roots in R (that is P (t) = 0 for all t ∈ R). However, by considering the collection of formal expressions a + bi [a, b ∈ R] with the obvious formal definitions of addition and multiplication and subject to the further condition i2 + 1 = 0, we obtain a field C ⊇ R in which P has a root (since P (i) = 0). We can perform a similar trick with other fields.

Iv) We continue with the notation (iii). The BCH code with {β, β 2} as defining set is Hamming’s original (7,4) code. The next theorem contains the key fact about BCH codes. 15. The minimum distance for a BCH code is at least as great as the design distance. 12 is of full rank for a BCH. To do this we use a result which every undergraduate knew in 1950. 16. [The van der Monde determinant] We work over a field K. The determinant 1 x1 x21 .. 1 x2 x22 .. 1 x3 x23 .. x1n−1 x2n−1 x3n−1 ... ... .

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