A Minimum Distance Bound for 1-Generator Quasi-Cyclic Codes

Let Fq be the finite field of q elements and A=Fq[X]/(Xm-1) be the algebra of q-ary polynomials modulo Xm-1. The q-ary 1-generator quasi-cyclic (QC) code of block length ml, of index a divisor of l, with generator a͟(X) = (ai(X))i=0l-1 is the A-cyclic submodule V of Al defined as Aa͟(X) = {(λ(X)ai(X))0l-1, λ(X) ϵ A} under the module operation λ(X)Σi=0l-1ai(X)Yi = Σi=0l-1㎻(X)ai(X)Yi, where ㎻(X) ϵ A and ㎻(X)ai(X) is reduced modulo Xm-1. Under the simplifying assumption (m, q) = 1, we determine a lower bound on the minimum distance of V. The result is achieved by determining a lower bound on the minimal number of not null blocks in a typical codeword of V.