TY - JOUR
T1 - An infinite family of Griesmer quasi-cyclic self-orthogonal codes
AU - Kim, Bohyun
AU - Lee, Yoonjin
AU - Yoo, Jinjoo
N1 - Publisher Copyright:
© 2021
PY - 2021/12
Y1 - 2021/12
N2 - Our aim for this paper is to find the construction method for quasi-cyclic self-orthogonal codes over the finite field Fpm. We first explicitly determine the generators of α-constacyclic codes over the finite Frobenius non-chain ring Rp,m=Fpm[u,v]/〈u2=v2=0,uv=vu〉, where m is a positive integer, α=a+ub+vc+uvd is a unit of Rp,m, a,b,c,d∈Fpm, and a is nonzero. We then find a Gray map from Rp,m[x]/〈xn−α〉 (with respect to homogeneous weights) to Fpm[x]/〈xp3m+1n−a〉 (with respect to Hamming weights), which is linear and preserves minimum weights. We present an efficient algorithm for finding the Gray images of α-constacyclic codes over Rp,m of length n, which produces infinitely many quasi-cyclic self-orthogonal codes over Fpm of length p3m+1 and index p3m. In particular, some family turns out to be “Griesmer” codes; these Griesmer quasi-cyclic self-orthogonal codes are “new” codes compared with previously known Griesmer codes of dimension 4.
AB - Our aim for this paper is to find the construction method for quasi-cyclic self-orthogonal codes over the finite field Fpm. We first explicitly determine the generators of α-constacyclic codes over the finite Frobenius non-chain ring Rp,m=Fpm[u,v]/〈u2=v2=0,uv=vu〉, where m is a positive integer, α=a+ub+vc+uvd is a unit of Rp,m, a,b,c,d∈Fpm, and a is nonzero. We then find a Gray map from Rp,m[x]/〈xn−α〉 (with respect to homogeneous weights) to Fpm[x]/〈xp3m+1n−a〉 (with respect to Hamming weights), which is linear and preserves minimum weights. We present an efficient algorithm for finding the Gray images of α-constacyclic codes over Rp,m of length n, which produces infinitely many quasi-cyclic self-orthogonal codes over Fpm of length p3m+1 and index p3m. In particular, some family turns out to be “Griesmer” codes; these Griesmer quasi-cyclic self-orthogonal codes are “new” codes compared with previously known Griesmer codes of dimension 4.
KW - Gray map
KW - Griesmer code
KW - Quasi-cyclic code
KW - Self-orthogonal code
UR - http://www.scopus.com/inward/record.url?scp=85114828924&partnerID=8YFLogxK
U2 - 10.1016/j.ffa.2021.101923
DO - 10.1016/j.ffa.2021.101923
M3 - Article
AN - SCOPUS:85114828924
SN - 1071-5797
VL - 76
JO - Finite Fields and their Applications
JF - Finite Fields and their Applications
M1 - 101923
ER -