Get An Introduction to Algebraic and Combinatorial Coding Theory PDF

By Ian F. Blake

ISBN-10: 0121035603

ISBN-13: 9780121035600

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Extra info for An Introduction to Algebraic and Combinatorial Coding Theory

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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. The work tends to be on factorizing polynomials and determining irreducible polynomials. The books by Dickson (1958) and Albert (1956) contain much information on these problems. We have relegated to the exercises some of these interesting results. On the factorization of polynomials, the algorithm by Berlekamp is very efficient and is described in Chapter 6 of his book (Berlekamp, 1968).

Conversely, if tr y φ 0, then there can be no solution in GF(q). There are many other constructions of irreducible polynomials and the reader is referred to the books by Dickson (1958, pp. 19-44) and Albert (1956, pp. 132-155). A few of the more interesting theorems have been included as exercises at the end of this chapter. 1J Cyclic Codes The class of linear codes is unfortunately still too general for us to say much on the problem of constructing codes with good distance properties. We therefore add another restriction.

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.

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


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