Constructions of Polynomial Codes Based on Circular Permutation Over Finite Fields

As a kind of very important error-correcting codes, polynomial codes have the advantages of good performance and simple structure. Generalized R-S codes over
$F_{q}$
as polynomial codes have the advantages described above, but their lengths
$n$
are at most
$q$
. In order to increase the length of the codes, in this paper, we apply a method similar to the construction of the generalized R-S codes over
$F_{q}$
and develop Ling and Xing’s idea to construct new
$q$
-ary linear codes. First, based on the idea of constructing polynomials codes over
$F_{q}$
, we construct a class of polynomial functions by using the circular permutation defined over
$F_{q}$
, and a new type of linear codes is constructed from the properties of these functions and Mobius functions. Then, we give the value of the length
$n$
, the dimension
$k$
and the lower bound of the minimum distance
$d$
of the new linear codes. It turns out that some codes from this construction have good parameters based on Brouwer’s table.