A Lower Bound on the Minimum Distance of a 1-Generator Quasi-Cyclic Code

Let F_q be the finite field of q elements and A=F_q [X]/(Xⁿ-1) be the algebra of q-ary polynomials modulo X ⁿ-1. The 1-generator quasi-cyclic (QC) code of block length nm over F_q, of index a divisor of m, with generator alowbar(X)=(a_i(X))₀ ^m-1 is the A-cyclic submodule of A^m defined as Aalowbar(X)={(lambda(X)a_i(X))₀ ^m-1 ,lambda(X)isin A}, under the module operation lambda(X)Sigma_i=0 ^ma_i(X)Yⁱ =Sigma_i=0 ^m-1lambda(X)a_i(X)Y ⁱ lambda(X)isinA, (a₀(X),a₁(X),..., a_m-1(X))isinA^m, where lambda(X)a_i(X) is reduced modulo Xⁿ-1. Assuming that g.c.d(n,q)=1, we show that the projections of a q-ary 1-generator QC code V according to its components are q-ary cyclic codes. Based on this property, we determine a lower bound on the minimum distance of a 1-generator QC code by means of this generator