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    Coding Theory : Cyclic Codes

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    4 (i) Let C be a linear code in IF. Explain what is meant when we say that C is cyclic. Give also the algebraic characterisation of cyclic codes using the ring
    (ii) Explain why the cyclic codes in R are in 1-1 correspondence with the monic polynomials in IFq[xJ that divide ? 1. Give the definition of the generator polynomial of a cyclic code in R.
    (iii) Let g E lFq[x] be the generator polynomial of a cyclic code C c R. Show that
    is a basis of C, where r is the degree of g and, for f E Fq[xJ, f denotes the residue class of f in R.
    (iv) Let C and g be as in part (ii). Find a generator matrix of C in terms of the coefficients of g.
    (v) Given that
    in IF2 [x], write down a generator matrix-and & parity check matrix fo a binary cyclic code of length 7 and dimension 4.
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    There is an isomorphism:

    Therefore a codeword c can be identified with a polynomial c(t) and we can think of a cyclic code as a subset of We want to prove that cyclic codes correspond to ideals. The fact that the sum of polynomials corresponding to codewords corresponds to ...

    Solution Summary

    Cyclic codes are investigated.