A Short Course on Error Correcting Codes (CISM International by N.J.A. Sloane

By N.J.A. Sloane

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Koenig. Preuss. Akad. , pp. 1152-1156. [ 89] Weyl, H. J. 1 Introducdon Cyclic codes are the most studied of all codes. After giving some of the general theory in the first few sections, we proceed with a brief description of the Hamming, BCH, Reed-Solomon, and Justesen codes, with a digression on perfect codes. 2 Definition of Cyclic Code is cyclic if it is linear and if any cyclic shift of a ( 5. 1) Definition A code codeward is also a codeword. , if (ao ,a 1 , ... ,an_ 1 ) E CC then (an- 1 , ao ,a 1 , ...

J. 1 Introducdon Cyclic codes are the most studied of all codes. After giving some of the general theory in the first few sections, we proceed with a brief description of the Hamming, BCH, Reed-Solomon, and Justesen codes, with a digression on perfect codes. 2 Definition of Cyclic Code is cyclic if it is linear and if any cyclic shift of a ( 5. 1) Definition A code codeward is also a codeword. , if (ao ,a 1 , ... ,an_ 1 ) E CC then (an- 1 , ao ,a 1 , ... ,an_ 2 )€ CC. 5) are cyclic codes. To get an algebraic description, we associate with the vector a = (ao,a 1 , ...

Example XIS + 1 = M(O)(x) M(l) (x) M (O) (x) M (l) (x) M ( 3 ) (x) M(S)(x) M (7 ) (x) = = = M( 3 )(x) M(S)(x) M( 7 )(x), X + 1, x4+ X+ 1, x4+ x3+ x2+ X + 1, = x2+ = x4+ X + 1, x3+ 1. 5) occurs when n = 2m -1. Then we have x2 m + x = product of all distinct irreducible polynomials over GF(2) whose degree divides m. 6) Example x 2+ x = x(x x 4+ x = x8 + x = x(x + 1) x(x + 1) (x2 + x +1) + 1)(x3 + x + 1)(x3 + x 2 + 1). ;;; 16. 8) Show that if s is relatiyely prime to n, then C8 contains m elements.

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