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]/(X
ⁿ
-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

^m
a
_i
(X)Y
ⁱ
=Sigma
_i=0

^m-1
lambda(X)a
_i
(X)Y
ⁱ
lambda(X)isinA, (a
₀
(X),a
₁
(X),..., a
_m-1
(X))isinA
^m
, where lambda(X)a
_i
(X) is reduced modulo X
ⁿ
-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