Skew-cyclic codes

Abstract

International audienceWe generalize the notion of cyclic codes by using generator polynomials in (non commutative) skew polynomial rings. Since skew polynomial rings are left and right euclidean, the obtained codes share most properties of cyclic codes. Since there are much more skew-cyclic codes, this new class of codes allows to systematically search for codes with good properties. We give many examples of codes which improve the previously best known linear codes

Skew cyclic codes of arbitrary length

“…It is shown in [2,3] that under the usual identification of vectors with polynomials, the skew cyclic shift (the second condition in the definition above) corresponds to skew multiplication by x in F [x; θ], and skew cyclic codes are ideals in F [x; θ]/(x n − 1) when m|n.…”

“…Boucher, W. Geiselmann and F. Ulmer in [2], and in [3], took another direction when they studied a more generalized class of linear and cyclic codes using a non-commutative ring. They studied what they called skew cyclic codes, where the generator polynomial of a skew cyclic code comes from a non-commutative ring F [x; θ], where F is a finite field and θ is a field automorphism of F .…”

“…They studied what they called skew cyclic codes, where the generator polynomial of a skew cyclic code comes from a non-commutative ring F [x; θ], where F is a finite field and θ is a field automorphism of F . They gave some examples of skew cyclic codes with Hamming distances larger than the best known linear codes with the same parameters [2].…”

“…The skew cyclic codes constructed in [2] and skew QC constructed in [1] have the property that | θ | = m must be a factor of the length n of the code. If m n then the polynomial generated by (x n − 1) is not a two-sided ideal and hence the set R n = F [x; θ]/(x n − 1) fails to be a ring.…”

Skew Cyclic Codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^{2}\mathbb{F}_{q}$

“…In [8], Hammons et al showed that some important binary nonlinear codes can be obtained from cyclic codes over Z 4 through the Gray map. Recently, in [2], D. Boucher et al introduced the class of θ-cyclic (skew cyclic) codes that generalizes the concept of cyclic codes over non-commutative polynomial rings, called a skew polynomial ring, to construct these types of codes.…”