An Introduction to Algebraic and Combinatorial Coding Theory by Ian F. Blake

By Ian F. Blake

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1 into sets st where each set is invariant under the mapping tf(n)(0 = iq mod n To each set s{;= {j\ jq, .. ,jqk~1}, integers mod «, there corresponds an irreducible polynomial with zeros η\ η]ρ, . . , η^~\ exponents mod n. The collection of these sets will be denoted by Σ ^ ) and the cardinality of this set aq{n). A set s, e Σ ^ ) will be called a q chain of n or a q chain mod n. ), then the polynomial is a polynomial over GF(q) and is a factor of xn — 1. Furthermore, any factor of x" — 1 is formed in this manner.

In this section we develop the results of MacWilliams and Pless relating the weight structure of a code to that of its dual. /2, then it may be easier to determine the weight structure of the dual code, which contains fewer vectors. We turn now to the formulas of MacWilliams and Pless. We define the weight enumerator of a code to be A(z)= ΣΑ,ζ' i=0 if the code contains Av codewords of weight /. Following the notation of MacWilliams (1962), we will denote the weight enumerator of a given code # by A(z) and that of its dual ^ 1 by B(z).

The dimension of the code, is n — mr = n — m deg g(z). Thus an (L, g) code has length \L\, dimension ^ [Λ - w deg #(z)], and distance d ^ 2 deg #(z) + 1 if #(z) has distinct roots. There will doubtless be much investigation of these codes. The minimum distance of a linear code is the weight of the minimum weight codeword, and this is a critical parameter in the evaluation of a code. However, in many practical instances it is important to know the number of codewords of each weight, and this determination is labeled the weight enumeration problem.

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