By Ian F. Blake

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**Extra resources for An Introduction to Algebraic and Combinatorial Coding Theory**

**Sample text**

Clearly H is a parity check matrix of a Hamming code with appropriate parameters. If H is considered as the generator matrix of a code #, then we claim that every nonzero codeword in <& has weight qk ~ ! , α , - 2 Χ) the corresponding word in the maximum length code. If w is the weight of X then (q — l)w = (q - l)qk~\ from which the result follows. Thus the weight enumerator of the code is A(z)= £ Λ ( ζ' = 1 + ( < ? * - l ^ " ' i=0 Using the MacWilliams identity B{z) = ί Bt z' = 1 £ [1 + (g - l)z]' "'(I - z)'· i=0 4 i=0 it readily follows that \)z)1 + (qk - 1)[1 + (?

Otherwise, we call it a nonprimitive BCH code. This terminology will carry over to other codes as well. Binary BCH codes have 52 1 Finite Fields and Coding Theory received a considerable amount of attention and a great deal more is known about specific cases than the foregoing outline would indicate. The reader is referred to Peterson and Weldon (1972) and Berlekamp (1968). 5. 2) ofthat section the (L, g) code is defined as the null space of the matrix g '(«i) 1 a, H= v r-l 9~\*n) If we choose the set L = {a1?

Consider the polynomial g(x) defined as gM= Now [gWY =Ϋ\(χi=0 Π (*-**') i=0 «qi)q = Π (** - «*'+ ' ) = Π (** - «'') = 9(xq) i=0 i=0 and this implies that, if g(x) = ^' l = 0 M*> then ι=0 ι=0 whence b? = b{ and g(x) e GF(q)[x]. Since g(x)\f(x) in the splitting field for /(*), then g(x)\f(x) in GFi^)^] and hence g(x) =f(x). It follows that k = I. Notice that the elements 1, a, a2, . . ,