A Course in Algebraic Error-Correcting Codes by Simeon Ball
Author:Simeon Ball
Language: eng
Format: epub, pdf
ISBN: 9783030411534
Publisher: Springer International Publishing
To construct a generator matrix (g ij) for the Reed–Solomon code, we choose k linearly independent polynomials f 1(X), …, f k(X) of degree at most k − 1 and index the rows with these polynomials. Then we index the columns with the elements a 1, …, a q of . The entry g ij = f i(a j) for and the g i,q+1 entry is the coefficient of X k−1 in the polynomial f i(X).
For example, with f i(X) = X i−1 the matrix
is a generator matrix for the Reed–Solomon code.
What makes Reed–Solomon codes so attractive for implementation is the availability of fast decoding algorithms. In the following theorem, we prove that there is a decoding algorithm that will correct up to t = ⌊(d − 1)∕2⌋ errors, where d is the minimum distance.
Although it is not really necessary, to make the proof of the following theorem easier, we shall only use the shortened Reed–Solomon code in which we delete the last coordinate. In this way every coordinate of a codeword is the evaluation of a polynomial at an element of .
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