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

Let F_q be the finite field of q elements and A=F_q[X]/(X^m-1) be the algebra of q-ary polynomials modulo X^m-1. The q-ary 1-generator quasi-cyclic (QC) code of block length ml, of index a divisor of l, with generator a(X) = (a_i(X))_i=0 ^l-1 is the A-cyclic submodule V of A^l defined as Aa(X) = {(lambda(X)a_i(X))₀ ^l-1, lambda(X) isin A} under the module operation lambda(X) Sigma_i=0 ^l-1 a_i(X)Yⁱ = Sigma_i=0 ^l-1lambda(X)a_i(X)Yⁱ, where lambda(X) isin A and lambda(X)a_i(X) is reduced modulo X^m-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.