The Subfield Codes of Some [q + 1, 2, q] MDS Codes

Recently, subfield codes of geometric codes over large finite fields {\mathrm {GF}}(q) with dimension 3 and 4 were studied and distance-optimal subfield codes over {\mathrm {GF}}(p) were obtained, where q=p^{m} . The key idea for obtaining very good subfield codes over small fields is to choose...

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Bibliographic Details
Published in:IEEE transactions on information theory 2022-06, Vol.68 (6), p.3643-3656
Main Authors: Heng, Ziling, Ding, Cunsheng
Format: Article
Language:English
Subjects:
Additives
Binary system
Codecs
Codes
Fields (mathematics)
Generators
Hamming weight
Indexes
Linear code
Linear codes
oval polynomial
subfield code
weight distribution
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Summary:Recently, subfield codes of geometric codes over large finite fields {\mathrm {GF}}(q) with dimension 3 and 4 were studied and distance-optimal subfield codes over {\mathrm {GF}}(p) were obtained, where q=p^{m} . The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of [q+1, 2, q] MDS codes over {\mathrm {GF}}(q) , and then studies the subfield codes over {\mathrm {GF}}(p) of some of the [q+1, 2,q] MDS codes over {\mathrm {GF}}(q) . Two families of dimension-optimal codes over {\mathrm {GF}}(p) are obtained, and several families of nearly optimal codes over {\mathrm {GF}}(p) are produced. Several open problems are also proposed in this paper.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2022.3151721