Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 363Additional Exercises: Error Correction for BCH CodesBCH codes have very attractive error correction algorithms. Let C be a BCH codein R n , and suppose that a code polynomial c(t) = c 0 + c 1 t + · · · + c n−1 t n−1 istransmitted. Let w(t) = w 0 + w 1 t + · · · w n−1 t n−1 be the polynomial in R n that isreceived. If errors have occurred in bits a 1 , . . . , a k , then w(t) = c(t) + e(t), wheree(t) = t a1 +t a2 +· · ·+t a kis the error polynomial. The decoder must determinethe integers a i and then recover c(t) from w(t) by flipping the a i th bit. From w(t)we can compute w(ω i ) = s i for i = 1, . . . , 2r, where ω is a primitive nth root ofunity over Z 2 . We say the syndrome of w(t) is s 1 , . . . , s 2r .1. Show that w(t) is a code polynomial if and only if s i = 0 for all i.2. Show thats i = w(ω i ) = e(ω i ) = ω ia1 + ω ia2 + · · · + ω ia kfor i = 1, . . . , 2r. The error-locator polynomial is defined to bes(x) = (x + ω a1 )(x + ω a2 ) · · · (x + ω a k).3. Recall the (15, 7)-block BCH code in Example 7. By Theorem 7.3, this codeis capable of correcting two errors. Suppose that these errors occur in bitsa 1 and a 2 . The error-locator polynomial is s(x) = (x + ω a1 )(x + ω a2 ). Showthat(s(x) = x 2 + s 1 x + s 2 1 + s )3.s 14. Let w(t) = 1 + t 2 + t 4 + t 5 + t 7 + t 12 + t 13 . Determine what the originallytransmitted code polynomial was.References and Suggested Readings[1] Childs, L. A Concrete Introduction to Higher Algebra. Springer-Verlag, NewYork, 1979.[2] Gåding, L. and Tambour, T. Algebra for Computer Science. Springer-Verlag,New York, 1988.[3] Lidl, R. and Pilz, G. Applied Abstract Algebra. Springer-Verlag, New York,1984. An excellent presentation of finite fields and their applications.[4] Mackiw, G. Applications of Abstract Algebra. Wiley, New York, 1985.[5] Roman, S. Coding and Information Theory. Springer-Verlag, New York,1992.[6] van Lint, J. H. Introduction to Coding Theory. Springer-Verlag, New York,1982.