Some optimal cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^t} $-codes

Yun Gao; Shilin Yang; Fang-Wei Fu

doi:10.3934/amc.2020072

Article Contents

2021, Volume 15, Issue 3: 387-396. Doi: 10.3934/amc.2020072

This issue Previous Article Next Article Ironwood meta key agreement and authentication protocol

Some optimal cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^t} $-codes

1.
College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
2.
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China

^* Corresponding author: Yun Gao
^* Corresponding author: Yun Gao

Received: September 2019

Revised: December 2019

Early access: April 2020

Published: August 2021

The authors would like to thank the anonymous reviewers and the Associate Editor for their valuable suggestions and comments that helped to greatly improve the paper. This research is supported by the 973 Program of China (Grant No. 2013CB834204), the National Natural Science Foundation of China (Grant No. 11671024, 61571243), and the Fundamental Research Funds for the Central Universities of China.

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Abstract

Let $ \mathbb{F}_{q^t} $ be a finite field of cardinality $ q^t $, where $ q $ is a power of a prime number $ p $ and $ t\geq 1 $ is a positive integer. Firstly, a family of cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^t} $-codes of length $ n $ is given, where $ n $ is a positive integer coprime to $ q $. Then according to the structure of this kind of codes, we construct $ 60 $ optimal cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^2} $-codes which have the same parameters as the MDS codes over $ \mathbb{F}_{q^2} $.

Keywords:

Cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^t} $-codes,
optimal codes,
MDS codes.

Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 11T71.

Citation:

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Table 1. Some optimal cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^2} $-codes of length $ n $

$ \{q,n\} $	Basis	Hamming weight enumerator	$ (n,(q^2)^k,d) $
$ \{2,5\} $	$ \alpha_1 $	$ W_1 $	$ (5,(2^2)^2,4) $
$ \{3,5\} $	$ \alpha_2 $	$ W_2 $	$ (5,(3^2)^2,4) $
$ \{3,7\} $	$ \alpha_3 $	$ W_3 $	$ (7,(3^2)^3,5) $
$ \{5,7\} $	$ \alpha_{4} $	$ W_{4} $	$ (7,(5^2)^3,5) $
$ \{5,13\} $	$ \alpha_{5} $	$ W_{5} $	$ (13,(5^2)^2,12) $
$ \{5,17\} $	$ \alpha_{6} $	$ W_{6} $	$ (17,(5^2)^8,10) $
$ \{7,5\} $	$ \alpha_{7} $	$ W_{7} $	$ (5,(7^2)^2,4) $
$ \{7,11\} $	$ \alpha_{8} $	$ W_{8} $	$ (11,(7^2)^5,7) $
$ \{7,13\} $	$ \alpha_{9} $	$ W_{9} $	$ (13,(7^2)^6,8) $
$ \{11,13\} $	$ \alpha_{10} $	$ W_{10} $	$ (13,(11^2)^6,8) $
$ \{11,17\} $	$ \alpha_{11} $	$ W_{11} $	$ (17,(11^2)^8,10) $
$ \{13,5\} $	$ \alpha_{12} $	$ W_{12} $	$ (5,(13^2)^2,4) $
$ \{13,11\} $	$ \alpha_{13} $	$ W_{13} $	$ (11,(13^2)^5,7) $
$ \{13,17\} $	$ \alpha_{14} $	$ W_{14} $	$ (17,(13^2)^2,16) $
$ \{13,19\} $	$ \alpha_{15} $	$ W_{15} $	$ (19,(13^2)^9,11) $
$ \{17,5\} $	$ \alpha_{16} $	$ W_{16} $	$ (5,(17^2)^2,4) $
$ \{17,7\} $	$ \alpha_{17} $	$ W_{17} $	$ (7,(17^2)^3,5) $
$ \{17,11\} $	$ \alpha_{18} $	$ W_{18} $	$ (11,(17^2)^5,7) $
$ \{17,13\} $	$ \alpha_{19} $	$ W_{19} $	$ (13,(17^2)^3,11) $
$ \{19,7\} $	$ \alpha_{20} $	$ W_{20} $	$ (7,(19^2)^3,5) $
$ \{19,11\} $	$ \alpha_{21} $	$ W_{21} $	$ (11,(19^2)^5,7) $
$ \{19,13\} $	$ \alpha_{22} $	$ W_{22} $	$ (13,(19^2)^6,8) $
$ \{19,17\} $	$ \alpha_{23} $	$ W_{23} $	$ (17,(19^2)^4,14) $
$ \{23,5\} $	$ \alpha_{24} $	$ W_{24} $	$ (5,(23^2)^2,4) $
$ \{23,13\} $	$ \alpha_{25} $	$ W_{25} $	$ (13,(23^2)^3,11) $
$ \{23,17\} $	$ \alpha_{26} $	$ W_{26} $	$ (17,(23^2)^8,10) $
$ \{29,11\} $	$ \alpha_{27} $	$ W_{27} $	$ (11,(29^2)^5,7) $
$ \{29,17\} $	$ \alpha_{28} $	$ W_{28} $	$ (17,(29^2)^8,10) $
$ \{31,7\} $	$ \alpha_{29} $	$ W_{29} $	$ (7,(31^2)^3,5) $
$ \{31,13\} $	$ \alpha_{30} $	$ W_{30} $	$ (13,(31^2)^2,12) $
$ \{31,17\} $	$ \alpha_{31} $	$ W_{31} $	$ (17,(31^2)^8,10) $
$ \{37,5\} $	$ \alpha_{32} $	$ W_{32} $	$ (5,(37^2)^2,4) $
$ \{37,13\} $	$ \alpha_{33} $	$ W_{33} $	$ (13,(37^2)^6,8) $
$ \{41,11\} $	$ \alpha_{34} $	$ W_{34} $	$ (11,(41^2)^5,7) $
$ \{41,13\} $	$ \alpha_{35} $	$ W_{35} $	$ (13,(41^2)^6,8) $
$ \{43,5\} $	$ \alpha_{36} $	$ W_{36} $	$ (5,(43^2)^2,4) $
$ \{43,13\} $	$ \alpha_{37} $	$ W_{37} $	$ (13,(43^2)^3,11) $
$ \{43,17\} $	$ \alpha_{38} $	$ W_{38} $	$ (17,(43^2)^4,14) $
$ \{47,5\} $	$ \alpha_{39} $	$ W_{39} $	$ (5,(47^2)^2,4) $
$ \{47,7\} $	$ \alpha_{40} $	$ W_{40} $	$ (7,(47^2)^3,5) $
$ \{47,13\} $	$ \alpha_{41} $	$ W_{41} $	$ (13,(47^2)^2,12) $
$ \{47,17\} $	$ \alpha_{42} $	$ W_{42} $	$ (17,(47^2)^2,16) $
$ \{53,5\} $	$ \alpha_{43} $	$ W_{43} $	$ (5,(53^2)^2,4) $
$ \{53,17\} $	$ \alpha_{44} $	$ W_{44} $	$ (17,(53^2)^4,14) $
$ \{59,7\} $	$ \alpha_{45} $	$ W_{45} $	$ (7,(59^2)^3,5) $
$ \{59,13\} $	$ \alpha_{46} $	$ W_{46} $	$ (13,(59^2)^6,8) $
$ \{59,17\} $	$ \alpha_{47} $	$ W_{47} $	$ (17,(59^2)^4,14) $
$ \{61,7\} $	$ \alpha_{48} $	$ W_{48} $	$ (7,(61^2)^3,5) $
$ \{61,11\} $	$ \alpha_{49} $	$ W_{49} $	$ (11,(61^2)^5,7) $
$ \{67,5\} $	$ \alpha_{50} $	$ W_{50} $	$ (5,(67^2)^2,4) $
$ \{67,13\} $	$ \alpha_{51} $	$ W_{51} $	$ (13,(61^2)^6,8) $
$ \{71,13\} $	$ \alpha_{52} $	$ W_{52} $	$ (13,(71^2)^6,8) $
$ \{73,11\} $	$ \alpha_{53} $	$ W_{53} $	$ (11,(73^2)^5,7) $
$ \{73,13\} $	$ \alpha_{54} $	$ W_{54} $	$ (13,(73^2)^2,12) $
$ \{79,11\} $	$ \alpha_{55} $	$ W_{55} $	$ (11,(79^2)^5,7) $
$ \{83,5\} $	$ \alpha_{56} $	$ W_{56} $	$ (5,(83^2)^2,4) $
$ \{83,11\} $	$ \alpha_{57} $	$ W_{57} $	$ (11,(83^2)^5,7) $
$ \{89,7\} $	$ \alpha_{58} $	$ W_{58} $	$ (7,(89^2)^3,5) $
$ \{89,17\} $	$ \alpha_{59} $	$ W_{59} $	$ (17,(89^2)^2,16) $
$ \{97,5\} $	$ \alpha_{60} $	$ W_{60} $	$ (5,(97^2)^2,4) $

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