American Institute of Mathematical Sciences

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

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

Yun Gao 1,, , Shilin Yang 1, and  Fang-Wei Fu 2,

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  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} $.

Keywords: Cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^t} $-codes, optimal codes, MDS codes.
Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 11T71.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

Y. L. Cao, X. 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.  Google Scholar

[7]

Y. L. Cao, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

R. Gabrys, E. Yaakobi, M. 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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

L. F. Jin, S. Ling, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

Y. L. Cao, X. 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.  Google Scholar

[7]

Y. L. Cao, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

R. Gabrys, E. Yaakobi, M. 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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

L. F. Jin, S. Ling, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

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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$ \{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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