# Coding Theory : Cyclic Codes

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

x7?1=(x+1)(x3+x+1)(x3+x2+1),

in IF2 [x], write down a generator matrix-and & parity check matrix fo a binary cyclic code of length 7 and dimension 4.

END OF PAPER

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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.