Constructing quasi-cyclic codes from linear algebra theory
Let F (q) be a finite field of cardinality q, l and m be positive integers and M (l) (F (q) ) the F (q) -algebra of all l x l matrices over F (q) . We investigate the relationship between monic factors of X (m) - 1 in the polynomial ring M (l) (F (q) )[X] and quasi-cyclic (QC) codes of length lm and index l over F (q) . Then we consider the idea of constructing QC codes from monic factors of X (m) - 1 in polynomial rings over F (q) -subalgebras of M (l) (F (q) ). This idea includes ideas of constructing QC codes of length lm and index l over F (q) from cyclic codes of length m over a finite field F (q) l, the finite chain ring F (q) + uF (q) + center dot center dot center dot A + u (l - 1) F (q) (u (l) = 0) and other type of finite chain rings.
quasi-cyclic code cyclic code polynomial ring over matrix ring reversible polynomial invariant subspace