An infinite family of Griesmer quasi-cyclic self-orthogonal codes

Bohyun Kim, Yoonjin Lee, Jinjoo Yoo

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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.

Original languageEnglish
Article number101923
JournalFinite Fields and their Applications
StatePublished - Dec 2021

Bibliographical note

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  • Gray map
  • Griesmer code
  • Quasi-cyclic code
  • Self-orthogonal code


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