Abstract Algebra Theory and Applications - Computer Science ...

358 CHAPTER 20 FINITE FIELDSand g(x) divides f(x),f(ω r ) = f(ω r+1 ) = · · · = f(ω r+s−1 ) = 0.Equivalently, we have the following system of equations:a i0 (ω r ) i 0+ a i1 (ω r ) i 1+ · · · + a is−1 (ω r ) i s−1= 0a i0 (ω r+1 ) i 0+ a i1 (ω r+1 ) i 2+ · · · + a is−1 (ω r+1 ) i s−1= 0a i0 (ω r+s−1 ) i 0+ a i1 (ω r+s−1 ) i 1+ · · · + a is−1 (ω r+s−1 ) i s−1= 0.Therefore, (a i0 , a i1 , . . . , a is−1 ) is a solution to the homogeneous system oflinear equations(ω i 0) r x 0 + (ω i 1) r x 1 + · · · + (ω i s−1) r x n−1 = 0(ω i 0) r+1 x 0 + (ω i 1) r+1 x 1 + · · · + (ω i s−1) r+1 x n−1 = 0(ω i 0) r+s−1 x 0 + (ω i 1) r+s−1 x 1 + · · · + (ω i s−1) r+s−1 x n−1 = 0.However, this system has a unique solution, since the determinant of thematrix ⎛(ω i 0) r (ω i 1) r · · · (ω i s−1) r ⎞(ω i 0) r+1 (ω i 1) r+1 · · · (ω i s−1) r+1⎜⎝... ..⎟. ⎠(ω i 0) r+s−1 (ω i 1) r+s−1 · · · (ω i s−1) r+s−1can be shown to be nonzero using Lemma 20.12 and the basic properties ofdeterminants (Exercise). Therefore, this solution must be a i0 = a i1 = · · · =a is−1 = 0.□BCH CodesSome of the most important codes, discovered independently by A. Hocquenghemin 1959 and by R. C. Bose and D. V. Ray-Chaudhuri in 1960, areBCH codes. The European and transatlantic communication systems bothuse BCH codes. Information words to be encoded are of length 231, anda polynomial of degree 24 is used to generate the code. Since 231 + 24 =255 = 2 8 − 1, we are dealing with a (255, 231)-block code. This BCH codewill detect six errors and has a failure rate of 1 in 16 million. One advantageof BCH codes is that efficient error correction algorithms exist for them...