Group structure on projective spaces and cyclic codes over finite fields
The finite field F\(_{q^n}\) can be used to describe the affine space AG\((n,q)\) of dimension \(n\) over the finite field F\(_q\), and the projective space PG\((n-1,q)\) can be described by associating with every point of PG\((n-1,q)\) exactly one element of F\(_{q^n}\setminus \{0\}\). Let \(F\) be a subspace of F\(_q[x_1,\ldots,x_n]\) and let \(S=\{s_1,\ldots,s_l\}\) be an indexed subset of F\(_{q^n}\). Then using \(F\) and \(S\), it is possible to define the Reed-Muller code RM\((F,S)\). The authors first of all study the geometrical properties of the subgroups of the multiplicative group of a finite extension F\(_{q^n}\) of F\(_q\) endowed with its vector space structure. It is shown that it is possible to find a subgroup \(G\) of the cyclic group on F\(_{q^n}^*\) inducing a bijective action on PG\((n-1,q)\) if and only if \(\gcd(n,q-1)=1\). This then implies that for \(\gcd(n,q-1) =1\), there are cyclic generalized projective Reed-Muller codes of order \(r\). The authors then study the connections between the Reed-Muller codes RM\((F,S)\) and the cyclic subgroups of the cyclic group on F\(_{q^n}^*\), and the article ends with a result on the parameters of a particular type of Reed-Muller codes RM\((F,S)\).
scientific article; zbMATH DE number 1471637