By Ian F. Blake
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Extra resources for An Introduction to Algebraic and Combinatorial Coding Theory
A set s, e Zq(qn — 1) will be called a q chain of qn — 1. A problem that is closely related to the foregoing, and one that we shall require for cyclic codes, is the factorization of xn — 1 over GF(q). Let GF(qm) be a splitting field for xn — 1 and let η be a primitive nth root of unity in GF(qm). Then if (AÏ, q) = 1, x" - 1 factors into the product of distinct minimal polynomials of the elements η\ i = 0, 1, . . , n — 1. ,η1*2, . . , η"** \ where k is the least positive integer such that η1*" = η' and the exponents are taken mod n.
We therefore add another restriction. 7 Cyclic Codes 41 Definition A code is cyclic if it is linear and if every cyclic shift of the coordinates of a codeword is a codeword. Thus if c = ( a 0 , a l 5 . . l9 α 0 , ο ^ , . . , a„_ 2 ) is also a codeword. It is convenient to identify a codeword with a codeword polynomial. The codeword polynomial for the codeword c is n-l i=0 A cyclic shift of this codeword c is equivalent to multiplication by x and reducing exponents mod n. However, reducing exponents mod n is equivalent to reducing the polynomial mod (xn — 1).
A little thought will show that if we let a be a primitive element of GF(2m), then we can as well express H as the 1 x (2m — 1) matrix H = [\ a a2 •••a 2 m " 2 ] where it is understood that a code vector is one whose corresponding polynomial has a as a zero. In this formulation the code is cyclic. 6, if h(x) is the primitive polynomial with root a, then the generator polynomial of the code is h(x). There are many ways of generalizing Hamming codes but we shall only discuss two of them. Define the parity check matrix H of a code over GF(q) as the m x [(qm — \)/(q — 1)] matrix with the property that no two of its columns are scalar multiples of one another.