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Article Contents

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

• * Corresponding author: Yun Gao

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

• 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 $\mathbb{F}_{q^2}$.

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

 Citation:

• 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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