TY - JOUR
T1 - On a class of skew constacyclic codes over mixed alphabets and applications in constructing optimal and quantum codes
AU - Dinh, Hai Q.
AU - Bag, Tushar
AU - Abdukhalikov, Kanat
AU - Pathak, Sachin
AU - Upadhyay, Ashish K.
AU - Bandi, Ramakrishna
AU - Chinnakum, Warattaya
N1 - Funding Information:
T. Bag and K. Abdukhalikov are supported by the UAEU Grant G00003490. A part of this paper was prepared when the T. Bag and S. Pathak were in IIT Patna, under financial support from the University Grant Commission (UGC), Govt. of India. A. K. Upadhyay is grateful to SERB DST Govt of India for financial support under MATRICS scheme with file no MTR/2020/000006. H. Q. Dinh and W. Chinnakum’s work have been partially supported by the Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Thailand. A part of this paper was written during a stay of H.Q. Dinh in the Vietnam Institute For Advanced Study in Mathematics (VIASM) in Summer 2022, he would like to thank the members of VIASM for their hospitality.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2023/2
Y1 - 2023/2
N2 - In this paper, we first discuss linear codes over R and present the decomposition structure of linear codes over the mixed alphabet FqR, where R= Fq+ uFq+ vFq+ uvFq, with u2 = 1,v2 = 1, uv = vu and q = pm for odd prime p, positive integer m. Let (Formula presented.) be an automorphism on Fq. Extending (Formula presented.) to (Formula presented.) over R, we study skew (Formula presented.)-(λ,Γ)-constacyclic codes over FqR, where λ and Γ are units in Fq and R, respectively. We also show that, the dual of a skew (Formula presented.)-(λ,Γ)-constacyclic code over FqR is a skew (Formula presented.)-(λ− 1,Γ− 1)-constacyclic code over FqR. We classify some self-dual skew (Formula presented.)-(λ,Γ)-constacyclic codes using the possible values of units of R. Also using suitable values of λ,(Formula presented.),Γ and (Formula presented.), we present the structure of other linear codes over FqR. We construct a Gray map over FqR and study the Gray images of skew (Formula presented.)-(λ,Γ)-constacyclic codes over FqR. As applications of our study, we construct many good codes, among them, there are 17 optimal codes and 2 near-optimal codes. Finally, we discuss the advantages in a construction of quantum error-correcting codes (QECCs) from skew (Formula presented.)-cyclic codes than from cyclic codes over Fq.
AB - In this paper, we first discuss linear codes over R and present the decomposition structure of linear codes over the mixed alphabet FqR, where R= Fq+ uFq+ vFq+ uvFq, with u2 = 1,v2 = 1, uv = vu and q = pm for odd prime p, positive integer m. Let (Formula presented.) be an automorphism on Fq. Extending (Formula presented.) to (Formula presented.) over R, we study skew (Formula presented.)-(λ,Γ)-constacyclic codes over FqR, where λ and Γ are units in Fq and R, respectively. We also show that, the dual of a skew (Formula presented.)-(λ,Γ)-constacyclic code over FqR is a skew (Formula presented.)-(λ− 1,Γ− 1)-constacyclic code over FqR. We classify some self-dual skew (Formula presented.)-(λ,Γ)-constacyclic codes using the possible values of units of R. Also using suitable values of λ,(Formula presented.),Γ and (Formula presented.), we present the structure of other linear codes over FqR. We construct a Gray map over FqR and study the Gray images of skew (Formula presented.)-(λ,Γ)-constacyclic codes over FqR. As applications of our study, we construct many good codes, among them, there are 17 optimal codes and 2 near-optimal codes. Finally, we discuss the advantages in a construction of quantum error-correcting codes (QECCs) from skew (Formula presented.)-cyclic codes than from cyclic codes over Fq.
KW - Gray map
KW - Mixed alphabet codes
KW - Optimal codes
KW - Skew (Formula presented.) − (λ,Γ)-constacyclic codes
UR - http://www.scopus.com/inward/record.url?scp=85133588619&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85133588619&partnerID=8YFLogxK
U2 - 10.1007/s12095-022-00594-3
DO - 10.1007/s12095-022-00594-3
M3 - Article
AN - SCOPUS:85133588619
SN - 1936-2447
VL - 15
SP - 171
EP - 198
JO - Cryptography and Communications
JF - Cryptography and Communications
IS - 1
ER -