Abstract Algebra Theory and Applications - Computer Science ...

354 CHAPTER 20 FINITE FIELDSThe additional ring structure on polynomial codes is very powerful indescribing cyclic codes. A cyclic shift of an n-tuple can be described bypolynomial multiplication. If f(t) = a 0 + a 1 t + · · · + a n−1 t n−1 is a codepolynomial in R n , thentf(t) = a n−1 + a 0 t + · · · + a n−2 t n−1is the cyclically shifted word obtained from multiplying f(t) by t. Thefollowing theorem gives a beautiful classification of cyclic codes in terms ofthe ideals of R n .Theorem 20.10 A linear code C in Z n 2in R n = Z[x]/〈x n − 1〉.is cyclic if and only if it is an idealProof. Let C be a linear cyclic code and suppose that f(t) is in C.Then tf(t) must also be in C. Consequently, t k f(t) is in C for all k ∈N. Since C is a linear code, any linear combination of the codewordsf(t), tf(t), t 2 f(t), . . . , t n−1 f(t) is also a codeword; therefore, for every polynomialp(t), p(t)f(t) is in C. Hence, C is an ideal.Conversely, let C be an ideal in Z 2 [x]/〈x n + 1〉. Suppose that f(t) =a 0 + a 1 t + · · · + a n−1 t n−1 is a codeword in C. Then tf(t) is a codeword inC; that is, (a 1 , . . . , a n−1 , a 0 ) is in C. □Theorem 20.10 tells us that knowing the ideals of R n is equivalent toknowing the linear cyclic codes in Z n 2 . Fortunately, the ideals in R n are easyto describe. The natural ring homomorphism φ : Z 2 [x] → R n defined byφ[f(x)] = f(t) is a surjective homomorphism. The kernel of φ is the idealgenerated by x n − 1. By Theorem 14.14, every ideal C in R n is of the formφ(I), where I is an ideal in Z 2 [x] that contains 〈x n − 1〉. By Theorem 15.12,we know that every ideal I in Z 2 [x] is a principal ideal, since Z 2 is a field.Therefore, I = 〈g(x)〉 for some unique monic polynomial in Z 2 [x]. Since〈x n − 1〉 is contained in I, it must be the case that g(x) divides x n − 1.Consequently, every ideal C in R n is of the formC = 〈g(t)〉 = {f(t)g(t) : f(t) ∈ R n and g(x) | (x n − 1) in Z 2 [x]}.The unique monic polynomial of the smallest degree that generates C iscalled the minimal generator polynomial of C.Example 6. If we factor x 7 − 1 into irreducible components, we havex 7 − 1 = (1 + x)(1 + x + x 3 )(1 + x 2 + x 3 ).