March 28, 2017

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By Ian F. Blake

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Delsarte (1973a) investigates codes over Abelian groups. Again, with the proper definition of a dual code, the MacWilliams identities still hold. There are also interesting relationships between Krawtchouk polynomials and weight enumeration [see MacWilliams et al. (1972c) and Delsarte (1973a)] and some of these are mentioned in Exercise 22. If the weight enumerator B(z) of <é>L is known, then the above theorem gives a set of n + 1 linear equations in the n + 1 unknowns A{,, i = 0, . . , n. In a practical situation it is possible that not all of the coefficients of B(z) may be known and in this case the so-called power moment identities of Pless (1963) may be important.

Together with the previous equations they are called the Newton identities. Apply these equations and the previous ones to the polynomial Ax)=tf-x which, of course, splits over GF(q) and has all the elements of GF(q) as zeros. l = — 1, we conclude that P{ = 0 for 0 < / < q — 1 and Pq_l = q — 1. 5 Let 0,, / > 0, denote the sum of the /th powers of the elements of GF(q). Then 0=[q-\ ' \0 if ( ? - D | / otherwise There is a fascinating literature on the study of polynomials over a finite field but it would be inappropriate to review it here.

Notice that for each set k\n. This process partitions elements into disjoint subsets. Elements of GF(q) are in sets of cardinality one. All other elements are in sets with more than one element. This partition of elements of GF(qn) induces a partition of the integers 0, 1, . . , qn — 2 where each set of integers comprises the exponents of the elements in a set in the partition of GF(qn). -i)(0 = '9niod(? n - 1) This mapping partitions the integers into disjoint sets which we shall denote by st and, again, the cardinality of each such set is a divisor of n.

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