Abstract Algebra Theory and Applications - Computer Science ...

20.2 POLYNOMIAL CODES 359The idea behind BCH codes is to choose a generator polynomial of smallestdegree that has the largest error detection and error correction capabilities.Let d = 2r + 1 for some r ≥ 0. Suppose that ω is a primitive nth rootof unity over Z 2 , and let m i (x) be the minimal polynomial over Z 2 of ω i . Ifg(x) = lcm[m 1 (x), m 2 (x), . . . , m 2r (x)],then the cyclic code 〈g(t)〉 in R n is called the BCH code of length n anddistance d. By Theorem 20.13, the minimum distance of C is at least d.Theorem 20.14 Let C = 〈g(t)〉 be a cyclic code in R n .statements are equivalent.The following1. The code C is a BCH code whose minimum distance is at least d.2. A code polynomial f(t) is in C if and only if f(ω i ) = 0 for 1 ≤ i < d.3. The matrix⎛H =⎜⎝1 ω ω 2 · · · ω n−1 ⎞1 ω 2 ω 4 · · · ω (n−1)(2)1 ω 3 ω 6 · · · ω (n−1)(3). ⎟. . . .. . ⎠1 ω 2r ω 4r · · · ω (n−1)(2r)is a parity-check matrix for C.Proof. (1) ⇒ (2). If f(t) is in C, then g(x) | f(x) in Z 2 [x]. Hence, fori = 1, . . . , 2r, f(ω i ) = 0 since g(ω i ) = 0. Conversely, suppose that f(ω i ) = 0for 1 ≤ i ≤ d. Then f(x) is divisible by each m i (x), since m i (x) is theminimal polynomial of ω i . Therefore, g(x) | f(x) by the definition of g(x).Consequently, f(x) is a codeword.(2) ⇒ (3). Let f(t) = a 0 + a 1 t + · · · + a n−1 vt n−1 be in R n . The correspondingn-tuple in Z n 2 is x = (a 0a 1 · · · a n−1 ) t . By (2),⎛Hx = ⎜⎝a 0 + a 1 ω + · · · + a n−1 ω n−1 ⎞ ⎛a 0 + a 1 ω 2 + · · · + a n−1 (ω 2 ) n−1⎟.⎠ = ⎜⎝a 0 + a 1 ω 2r + · · · + a n−1 (ω 2r ) n−1f(ω)f(ω 2 ).f(ω 2r )⎞⎟⎠ = 0exactly when f(t) is in C. Thus, H is a parity-check matrix for C.