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.
Bibliographical noteFunding Information:
Y. Lee is a corresponding author and supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) ( NRF-2017R1A2B2004574 ) and also by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177 ). J. Yoo is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A4A1016649 ) and also by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2021R1I1A1A01047765 ).
- Gray map
- Griesmer code
- Quasi-cyclic code
- Self-orthogonal code