August  2021, 15(3): 387-396. doi: 10.3934/amc.2020072

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

Received  September 2019 Revised  December 2019 Published  August 2021 Early access  April 2020

Fund Project: 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} $.

Citation: Yun Gao, Shilin Yang, Fang-Wei Fu. Some optimal cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^t} $-codes. Advances in Mathematics of Communications, 2021, 15 (3) : 387-396. doi: 10.3934/amc.2020072
References:
[1]

T. L. Alderson, Extending MDS codes, Ann. Comb., 9 (2005), 125-135.  doi: 10.1007/s00026-005-0245-7.

[2]

I. Bouyukliev and J. Simonis, Some new results on optimal codes over $\mathbb{F}_5$, Des. Codes Cryptogr., 30 (2003), 97-111.  doi: 10.1023/A:1024763410967.

[3]

S. Bouyuklieva and P. R. J. Östergảrd, New constructions of optimal self-dual binary codes of length 54, Des. Codes Cryptogr., 41 (2006), 101-109.  doi: 10.1007/s10623-006-0018-2.

[4]

W. BosmaJ. Cannon and C. Playoust, The Magma algebra system, J. Symb. Comput., 24 (1997), 235-265. 

[5]

Y. L. Cao and Y. Gao, Repeated root cyclic $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^l}$, Finite Fields Appl., 31 (2015), 202-227.  doi: 10.1016/j.ffa.2014.10.003.

[6]

Y. L. CaoX. X. Chang and Y. Cao, Constacyclic $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^l}$, Appl. Algebra Engrg. Comm. Comput., 26 (2015), 369-388.  doi: 10.1007/s00200-015-0257-4.

[7]

Y. L. CaoJ. Gao and F.-W. Fu, Semisimple multivariable $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^l}$, Des. Codes Cryptogr., 77 (2015), 153-177.  doi: 10.1007/s10623-014-9994-9.

[8]

B. C. Chen and H. W. Liu, New constructions of MDS codes with complementary duals, IEEE Trans. Inform. Theory, 64 (2018), 5776-5782.  doi: 10.1109/TIT.2017.2748955.

[9]

B. K. Dey and B. S. Rajan, $\mathbb{F}_q$-linear cyclic codes over $\mathbb{F}_{q^m}$: DFT approach, Des. Codes Cryptogr., 34 (2005), 89-116.  doi: 10.1007/s10623-003-4196-x.

[10]

S. Dodunekov and I. Landgev, On near-MDS codes, J. Geom., 54 (1995), 30-43.  doi: 10.1007/BF01222850.

[11]

R. GabrysE. YaakobiM. Blaum and P. H. Siegel, Constructions of partial MDS codes over small fields, IEEE Internat. Symposium Inform. Theory, 65 (2019), 3692-3701.  doi: 10.1109/TIT.2018.2890201.

[12]

M. Grassl and T. A. Gulliver, On self-dual MDS codes, IEEE Internat. Symposium Inform. Theory, (2008), 1954-1957. 

[13]

W. C. Huffman, Cyclic $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes, Int. J. Inf. and Coding Theory, 1 (2010), 249-284.  doi: 10.1504/IJICOT.2010.032543.

[14]

W. C. Huffman, Self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes with an automorphism of prime order, Adv. Math. Commun., 7 (2013), 57-90.  doi: 10.3934/amc.2013.7.57.

[15]

W. C. Huffman, On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes, Adv. Math. Commun., 7 (2013), 349-378.  doi: 10.3934/amc.2013.7.349.

[16]

B. Hurley and T. Hurley, Systems of MDS codes from units and idempotents, Discrete Math., 335 (2014), 81-91.  doi: 10.1016/j.disc.2014.07.010.

[17]

L. F. JinS. LingJ. Q. Luo and C. P. Xing, Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes, IEEE Trans. Inform. Theory, 56 (2010), 4735-4740.  doi: 10.1109/TIT.2010.2054174.

[18]

L. F. Jin and C. P. Xing, New MDS self-dual codes from generalized Reed-Solomon codes, IEEE Trans. Inform. Theory, 63 (2017), 1434-1438.  doi: 10.1109/TIT.2016.2645759.

[19]

T. Maruta, On the existence of cyclic and pseudo-cyclic MDS codes, Europ. J. Combinatorics, 19 (1998), 159-174.  doi: 10.1006/S0195-6698(97)90000-7.

[20]

R. M. Roth and G. Seroussi, On cyclic MDS codes of length $q$ over $GF(q)$, IEEE Trans. Inform. Theory, 32 (1986), 284-285.  doi: 10.1109/TIT.1986.1057151.

[21]

R. M. Roth and G. Seroussi, On generator matrices of MDS codes, IEEE Trans. Inform. Theory, 31 (1985), 826-830.  doi: 10.1109/TIT.1985.1057113.

[22]

M. J. Shi and P. Solé, Optimal $p$-ary codes from one-weight and two-weight codes over $\mathbb{F}_p+v{\mathbb{F}_p}^*$, J. Syst. Sci. Complex., 28 (2015), 679-690.  doi: 10.1007/s11424-015-3265-3.

[23]

Z.-X. Wan, Cyclic codes over Galois rings$^*$, Algebra Colloquium, 6 (1999), 291-304. 

show all references

References:
[1]

T. L. Alderson, Extending MDS codes, Ann. Comb., 9 (2005), 125-135.  doi: 10.1007/s00026-005-0245-7.

[2]

I. Bouyukliev and J. Simonis, Some new results on optimal codes over $\mathbb{F}_5$, Des. Codes Cryptogr., 30 (2003), 97-111.  doi: 10.1023/A:1024763410967.

[3]

S. Bouyuklieva and P. R. J. Östergảrd, New constructions of optimal self-dual binary codes of length 54, Des. Codes Cryptogr., 41 (2006), 101-109.  doi: 10.1007/s10623-006-0018-2.

[4]

W. BosmaJ. Cannon and C. Playoust, The Magma algebra system, J. Symb. Comput., 24 (1997), 235-265. 

[5]

Y. L. Cao and Y. Gao, Repeated root cyclic $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^l}$, Finite Fields Appl., 31 (2015), 202-227.  doi: 10.1016/j.ffa.2014.10.003.

[6]

Y. L. CaoX. X. Chang and Y. Cao, Constacyclic $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^l}$, Appl. Algebra Engrg. Comm. Comput., 26 (2015), 369-388.  doi: 10.1007/s00200-015-0257-4.

[7]

Y. L. CaoJ. Gao and F.-W. Fu, Semisimple multivariable $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^l}$, Des. Codes Cryptogr., 77 (2015), 153-177.  doi: 10.1007/s10623-014-9994-9.

[8]

B. C. Chen and H. W. Liu, New constructions of MDS codes with complementary duals, IEEE Trans. Inform. Theory, 64 (2018), 5776-5782.  doi: 10.1109/TIT.2017.2748955.

[9]

B. K. Dey and B. S. Rajan, $\mathbb{F}_q$-linear cyclic codes over $\mathbb{F}_{q^m}$: DFT approach, Des. Codes Cryptogr., 34 (2005), 89-116.  doi: 10.1007/s10623-003-4196-x.

[10]

S. Dodunekov and I. Landgev, On near-MDS codes, J. Geom., 54 (1995), 30-43.  doi: 10.1007/BF01222850.

[11]

R. GabrysE. YaakobiM. Blaum and P. H. Siegel, Constructions of partial MDS codes over small fields, IEEE Internat. Symposium Inform. Theory, 65 (2019), 3692-3701.  doi: 10.1109/TIT.2018.2890201.

[12]

M. Grassl and T. A. Gulliver, On self-dual MDS codes, IEEE Internat. Symposium Inform. Theory, (2008), 1954-1957. 

[13]

W. C. Huffman, Cyclic $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes, Int. J. Inf. and Coding Theory, 1 (2010), 249-284.  doi: 10.1504/IJICOT.2010.032543.

[14]

W. C. Huffman, Self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes with an automorphism of prime order, Adv. Math. Commun., 7 (2013), 57-90.  doi: 10.3934/amc.2013.7.57.

[15]

W. C. Huffman, On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes, Adv. Math. Commun., 7 (2013), 349-378.  doi: 10.3934/amc.2013.7.349.

[16]

B. Hurley and T. Hurley, Systems of MDS codes from units and idempotents, Discrete Math., 335 (2014), 81-91.  doi: 10.1016/j.disc.2014.07.010.

[17]

L. F. JinS. LingJ. Q. Luo and C. P. Xing, Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes, IEEE Trans. Inform. Theory, 56 (2010), 4735-4740.  doi: 10.1109/TIT.2010.2054174.

[18]

L. F. Jin and C. P. Xing, New MDS self-dual codes from generalized Reed-Solomon codes, IEEE Trans. Inform. Theory, 63 (2017), 1434-1438.  doi: 10.1109/TIT.2016.2645759.

[19]

T. Maruta, On the existence of cyclic and pseudo-cyclic MDS codes, Europ. J. Combinatorics, 19 (1998), 159-174.  doi: 10.1006/S0195-6698(97)90000-7.

[20]

R. M. Roth and G. Seroussi, On cyclic MDS codes of length $q$ over $GF(q)$, IEEE Trans. Inform. Theory, 32 (1986), 284-285.  doi: 10.1109/TIT.1986.1057151.

[21]

R. M. Roth and G. Seroussi, On generator matrices of MDS codes, IEEE Trans. Inform. Theory, 31 (1985), 826-830.  doi: 10.1109/TIT.1985.1057113.

[22]

M. J. Shi and P. Solé, Optimal $p$-ary codes from one-weight and two-weight codes over $\mathbb{F}_p+v{\mathbb{F}_p}^*$, J. Syst. Sci. Complex., 28 (2015), 679-690.  doi: 10.1007/s11424-015-3265-3.

[23]

Z.-X. Wan, Cyclic codes over Galois rings$^*$, Algebra Colloquium, 6 (1999), 291-304. 

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) $
$ \{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) $
[1]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349

[2]

W. Cary Huffman. Self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes with an automorphism of prime order. Advances in Mathematics of Communications, 2013, 7 (1) : 57-90. doi: 10.3934/amc.2013.7.57

[3]

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