Question: 4. (a) State the MacWilliams Duality Theorem for q-ary linear codes. (b) Show that if C, C2 is a pair of codes then the weight

4. (a) State the MacWilliams Duality Theorem for

4. (a) State the MacWilliams Duality Theorem for q-ary linear codes. (b) Show that if C, C2 is a pair of codes then the weight enumerator of their direct sum, C1 C2, is the product of the weight enumerators of C and C2- (c) Let C be an F2 [24, 12, 8) code satisfying C = C+, in which all words have weight divisible by 4. Show that C contains the all-one vector 1. Show that C has weight enumerator Ac(2) = 1 + az + b212 + a216 +224, for some nonnegative integers a, b. Use the MacWilliams Duality Theo- rem to show that a = 759, b = 2576. 4. (a) State the MacWilliams Duality Theorem for q-ary linear codes. (b) Show that if C, C2 is a pair of codes then the weight enumerator of their direct sum, C1 C2, is the product of the weight enumerators of C and C2- (c) Let C be an F2 [24, 12, 8) code satisfying C = C+, in which all words have weight divisible by 4. Show that C contains the all-one vector 1. Show that C has weight enumerator Ac(2) = 1 + az + b212 + a216 +224, for some nonnegative integers a, b. Use the MacWilliams Duality Theo- rem to show that a = 759, b = 2576

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