Skew Cyclic codes over $\F_q+u\F_q+v\F_q+uv\F_q$

Yao, Ting; Shi, Minjia; Solé, Patrick

doi:10.48550/arxiv.1504.07831

Cited by 1 publication

(3 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Example 5.4 In F 3 [x], x 10 − 1 = (x + 1)(x + 2)(x 4 + x 3 + x 2 + x + 1)(x 4 + 2x 3 + x 2 + 2x + 1), x 10 + 1 = (x 2 + 1)(x 4 + x 3 + 2x + 1)(x 4 + 2x 3 + x + 1). Let C 1 be the cyclic code generated by g 1 (x) = x + 1, and C 2 be the negative cyclic code generated by g 2 (x) = x 4 + x 3 + 2x + 1, then the parameters of C 1 , C 2 are [10, 9, 2], [10,6,4], and the parameters of η M (C 1 e 1 + C 2 e 2 ) are [20,15,4]. Look up the table [26] we know that this is an optimal linear code, and one of its generator matrices is…”

Section: Isomorphismmentioning

confidence: 99%

“…In fact, the optimal linear code with parameters [20,15,4] over F 3 in the table [26] is also constructed by (u|u + v) construction.…”

Section: Isomorphismmentioning

confidence: 99%

“…The ring F q [u, v]/(u 2 −u, v 2 −v) is isomorphic to the finite commutative semisimple ring F q ×F q ×F q ×F q , the study of linear codes over this ring is similar, Yao et al [15] studied skew cyclic codes over this ring, although they chose a different automorphism of this ring, they limited linear codes C over the ring satisfy specific conditions. Islam et al [16] also studied skew cyclic codes and skew constacyclic codes over this ring, the automorphism they chose can be classified into our special case one.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Skew constacyclic codes over a class of finite commutative semisimple rings

Zhao¹

2022

Preprint

View full text Add to dashboard Cite

Let F q be a finite field of q = p r elements, where p is a prime, r is a positive integer, we determine automorphism θ of a class of finite commutative semisimple ring R = t i=1 F q and the structure of its automorphism group Aut (R). We find that θ is totally determined by its action on the set of primitive idempotent e 1 , e 2 , . . . , e t of R and its action onwhere G 1 is a normal subgroup of Aut (R) isomorphic to the direct product of t cyclic groups of order r, and G 2 is a subgroup of Aut (R) isomorphic to the symmetric group S t of t elements.For any linear code C over R, we establish a one-to-one correspondence between C and t linear codes C 1 , C 2 , . . . , C t over F q by defining an isomorphism ϕ. For any θ in Aut (R) and any invertible element λ in R, we give a necessary and sufficient condition that a linear code over R is a θ-λ-cyclic code in a unified way. When θ ∈ G 1 , the C 1 , C 2 , . . . , C t corresponding to the θ-λ-cyclic code C over R are skew constacyclic codes over F q . When θ ∈ G 2 , the C 1 , C 2 , . . . , C t corresponding to the skew cyclic code C over R are quasi cyclic codes over F q . For general case, we give conditions that C 1 , C 2 , . . . , C t should satisfy when the corresponding linear code C over R is a skew constacyclic code.Linear codes over R are closely related to linear codes over F q . We define homomorphisms which map linear codes over R to matrix product codes over F q . One of the homomorphisms is a generalization of the ϕ used to decompose linear code over R into linear codes over F q , another homomorphism is surjective. Both of them can be written in the form of η M defined by us, but the matrix M used is different. As an application of the theory constructed above, we construct some optimal linear codes over F q .

show abstract