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A class of 1-generator quasi-cyclic codes

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1 Author(s)
Seguin, G.E. ; Dept. of Electr. & Comput. Eng., R. Mil. Coll., Kingston, Ont., Canada

If R = Fq[x]/(xm - 1), S = Fqn[x]/(xm - 1), we define the mapping a_(x) → A(x) =σ0n-1ai(x)αi from Rn onto S, where (α0, αi,..., αn-1) is a basis for Fqn over Fq. This carries the q-ray 1-generator quasicyclic (QC) code R a_(x) onto the code RA(x) in S whose parity-check polynomial (p.c.p.) is defined as the monic polynomial h(x) over Fq of least degree such that h(x)A(x) = 0. In the special case, where gcd(q, m) = 1 and where the prime factorizations of xm 1 over Fq and Fqn are the same we show that there exists a one-to-one correspondence between the q-ary 1-generator quasis-cyclic codes with p.c.p. h(x) and the elements of the factor group J* /I* where J is the ideal in S with p.c.p. h(x) and I the corresponding quantity in R. We then describe an algorithm for generating the elements of J*/I*. Next, we show that if we choose a normal basis for Fqn over Fq, then we can modify the aforementioned algorithm to eliminate a certain number of equivalent codes, thereby rending the algorithm more attractive from a computational point of view. Finally in Section IV, we show how to modify the above algorithm in order to generate all the binary self-dual 1-generator QC codes.

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Information Theory, IEEE Transactions on  (Volume:50 ,  Issue: 8 )