By Ian F. Blake
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Additional resources for An Introduction to Algebraic and Combinatorial Coding Theory
The algebra An is isomorphic to the group algebra GF(q)Cn, C„ the cyclic group of order w, and in this formulation it leads to an interesting generalization. The following theorem is of a fundamental nature. 1 An (w, A:) linear code # over GF(q) is cyclic iff it is an ideal of An. Proof Let <& be an ideal in An. It is clearly a linear subspace of An as a vector space and it remains to check that it is a cyclic subspace. But since # is an ideal, it is closed under multiplication by x and hence cyclic.
2, and the fact that xqn — x has simple zeros and can be factored into distinct, irreducible, monic polynomials. □ We now digress slightly to consider in some detail the above factorization and to give some notation that will be useful later. Recall the following definition. Definition The minimal polynomial of an element a e GF(qn) over GF(q) is the monic polynomial that is irreducible in GF(q)[x] and has a as a zero. 2 that any element a in GF(q") satisfies an irreducible polynomial over GF(q) of degree at most n.
Notice that for q = 2 the parameters of this code reduce to those of the binary code and the arguments remain valid and give the same code as in the binary case. That the above codes are perfect follows from the direct calculation qn = ^(«m-D/(4-i) = ( ? )/<«- l ) ] - m which is readily verified. By a straightforward argument, the weight enumerator polynomial for these codes can be obtained. As previously, we denote the number of words in the code of weight i by At and the weight enumerator polynomial by A(z)= £ Aj i=0 where n is the length of the code.
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