November  2020, 14(4): 555-572. doi: 10.3934/amc.2020029

Some results on $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes

1. 

School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, China

2. 

Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, China

3. 

School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410114, China

4. 

College of Science, Tianjin University of Science and Technology, Tianjin, 300071, China

* Corresponding author: Jian Gao, dezhougaojian@163.com

Received  December 2018 Revised  May 2019 Published  September 2019

Fund Project: This research is supported by the National Natural Science Foundation of China (Grant No. 11701336, 11626144 and 11671235), the Scientific Research Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Grant No. 2018MMAEZD09), the Scientific Research Fund of Hubei Provincial Key Laboratory of Applied Mathematics (Hubei University)(Grant No. AM201804.)

$ \mathbb{Z}_p\mathbb{Z}_p[v] $-Additive cyclic codes of length $ (\alpha,\beta) $ can be viewed as $ R[x] $-submodules of $ \mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1) $, where $ R = \mathbb{Z}_p+v\mathbb{Z}_p $ with $ v^2 = v $. In this paper, we determine the generator polynomials and the minimal generating sets of this family of codes as $ R[x] $-submodules of $ \mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1) $. We also determine the generator polynomials of the dual codes of $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Some optimal $ \mathbb{Z}_p\mathbb{Z}_p[v] $-linear codes and MDSS codes are obtained from $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Moreover, we also get some quantum codes from $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes.

Citation: Lingyu Diao, Jian Gao, Jiyong Lu. Some results on $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Advances in Mathematics of Communications, 2020, 14 (4) : 555-572. doi: 10.3934/amc.2020029
References:
[1]

T. AbualrubI. Siap and N. Aydin, $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes, IEEE Trans. Inform. Theory, 60 (2014), 1508-1514.  doi: 10.1109/TIT.2014.2299791.  Google Scholar

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T. Abualrub, I. Siap and I. Aydogdu, $\mathbb{Z}_2(\mathbb{Z}_2+u\mathbb{Z}_2)$-Linear Cyclic Codes, Proceedings of the International MultiConference of Engineers and Computer Scientists, Ⅱ, 2014. Google Scholar

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I. Aydogdu and I. Siap, On $\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive Codes, Linear Multilinear Algebra, 63 (2015), 2089-2102.  doi: 10.1080/03081087.2014.952728.  Google Scholar

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I. AydogduT. Abualrub and I. Siap, $\mathbb{Z}_2\mathbb{Z}_2[u]$-cyclic and constacyclic codes, IEEE Trans. Inform. Theory, 63 (2017), 4883-4893.  doi: 10.1109/TIT.2016.2632163.  Google Scholar

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J. Gao and Y. K. Wang, $u$-Constacyclic codes over $\mathbb{F}_p+u\mathbb{F}_p$ and their applications of constructing new non-binary quantum codes, Quantum Inf. Process, 17 (2018), Art. 4, 9 pp. doi: 10.1007/s11128-017-1775-8.  Google Scholar

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J. Gao and Y. K. Wang, Quantum codes derived from negacyclic codes, Int. J. Theor. Phys., 57 (2018), 682-686.  doi: 10.1007/s10773-017-3599-9.  Google Scholar

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Y. GaoJ. Gao and F.-W. Fu, Quantum codes from cyclic codes over the ring $\mathbb{F}_q + v_1\mathbb{F}_q +\cdots+ v_r\mathbb{F}_q$, Applicable Algebra in Engineering, Communication and Computing, 30 (2019), 161-174.  doi: 10.1007/s00200-018-0366-y.  Google Scholar

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M. E. Koroglu and I. Siap, Quantum codes from a class of constacyclic codes over group algebras, Malaysian Journal of Mathematical Sciences, 11 (2017), 289-301.   Google Scholar

[17]

F. H. Ma, J. Gao and F.-W. Fu, Constacyclic codes over the ring $\mathbb{F}_q+v\mathbb{F}_q+v_2\mathbb{F}_q$ and their applications of constructing new non-binary quantum codes, Quantum Inf. Processing, (2018), https://doi.org/10.1007/s11128-018-1898-6. Google Scholar

[18]

F. J. MacMilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[19]

R. C. Singleton, Maximum distance $q$-ary codes, IEEE Trans. Inform. Theory, 10 (1964), 116-118.  doi: 10.1109/tit.1964.1053661.  Google Scholar

[20]

B. Srinivasulu and M. Bhaintwal, $\mathbb{Z}_2(\mathbb{Z}_2+u\mathbb{Z}_2)$-Additive cyclic codes and their duals, Discrete Math. Algorithm. Appl., 8, (2016), 1650027, 19 pp. doi: 10.1142/S1793830916500270.  Google Scholar

[21]

Z.-X. Wan, Quaternary Codes, Series on Applied Mathematics, 8. World Scientific Publishing Co., Inc., River Edge, NJ, 1997. doi: 10.1142/9789812798121.  Google Scholar

[22]

S. X. ZhuY. Whang and M. J. Shi, Some results on cyclic codes over $\mathbb{F}_2+v\mathbb{F}_2$, IEEE Trans. Inform. Theory, 56 (2010), 1680-1684.  doi: 10.1109/TIT.2010.2040896.  Google Scholar

show all references

References:
[1]

T. AbualrubI. Siap and N. Aydin, $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes, IEEE Trans. Inform. Theory, 60 (2014), 1508-1514.  doi: 10.1109/TIT.2014.2299791.  Google Scholar

[2]

T. Abualrub, I. Siap and I. Aydogdu, $\mathbb{Z}_2(\mathbb{Z}_2+u\mathbb{Z}_2)$-Linear Cyclic Codes, Proceedings of the International MultiConference of Engineers and Computer Scientists, Ⅱ, 2014. Google Scholar

[3]

M. Ashraf and G. Mohammad, Construction of quantum codes from cyclic codes over $\mathbb{F}_p+v\mathbb{F}_p$, Int. J. Information and Coding Theory, 3 (2015), 137-144.  doi: 10.1504/IJICOT.2015.072627.  Google Scholar

[4]

I. AydogduT. Abualrub and I. Siap, On $\mathbb{Z}_2\mathbb{Z}_2[u]$-additive codes, Int. J. Comput. Math., 92 (2015), 1806-1814.  doi: 10.1080/00207160.2013.859854.  Google Scholar

[5]

I. Aydogdu and I. Siap, On $\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive Codes, Linear Multilinear Algebra, 63 (2015), 2089-2102.  doi: 10.1080/03081087.2014.952728.  Google Scholar

[6]

I. AydogduT. Abualrub and I. Siap, $\mathbb{Z}_2\mathbb{Z}_2[u]$-cyclic and constacyclic codes, IEEE Trans. Inform. Theory, 63 (2017), 4883-4893.  doi: 10.1109/TIT.2016.2632163.  Google Scholar

[7]

J. BorgesC. Fernández-CórdobaJ. Pujol and J. Rifà, $\mathbb{Z}_2\mathbb{Z}_4$-linear codes: Geneartor matrices and duality, Des. Codes Cryptogr., 54 (2010), 167-179.  doi: 10.1007/s10623-009-9316-9.  Google Scholar

[8]

J. BorgesC. Fernández-Córdoba and R. Ten-Valls, $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes, generator polynomials and dual codes, IEEE Trans. Inform. Theory, 62 (2016), 6348-6354.  doi: 10.1109/TIT.2016.2611528.  Google Scholar

[9]

A. R. CalderbankE. M. RainsP. M. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.  doi: 10.1109/18.681315.  Google Scholar

[10]

P. Delsarte and V. I. Levenshtein, Association schemes and coding theory: 1948–1998, IEEE Trans. Inform. Theory, 44 (1998), 2477-2504.  doi: 10.1109/18.720545.  Google Scholar

[11]

Y. Edel, Some Good Quantum Twisted Codes [Online], Available: https://www.mathi.uni-heidelberg.de/yves/Matritzen/QTBCH/QTBCHIndex.html. Google Scholar

[12]

J. Gao and Y. K. Wang, $u$-Constacyclic codes over $\mathbb{F}_p+u\mathbb{F}_p$ and their applications of constructing new non-binary quantum codes, Quantum Inf. Process, 17 (2018), Art. 4, 9 pp. doi: 10.1007/s11128-017-1775-8.  Google Scholar

[13]

J. Gao and Y. K. Wang, Quantum codes derived from negacyclic codes, Int. J. Theor. Phys., 57 (2018), 682-686.  doi: 10.1007/s10773-017-3599-9.  Google Scholar

[14]

Y. GaoJ. Gao and F.-W. Fu, Quantum codes from cyclic codes over the ring $\mathbb{F}_q + v_1\mathbb{F}_q +\cdots+ v_r\mathbb{F}_q$, Applicable Algebra in Engineering, Communication and Computing, 30 (2019), 161-174.  doi: 10.1007/s00200-018-0366-y.  Google Scholar

[15]

A. R. HammonsP. KumarA. R. CalderbankN. J. A. Sloane and P. Solé, The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154.  Google Scholar

[16]

M. E. Koroglu and I. Siap, Quantum codes from a class of constacyclic codes over group algebras, Malaysian Journal of Mathematical Sciences, 11 (2017), 289-301.   Google Scholar

[17]

F. H. Ma, J. Gao and F.-W. Fu, Constacyclic codes over the ring $\mathbb{F}_q+v\mathbb{F}_q+v_2\mathbb{F}_q$ and their applications of constructing new non-binary quantum codes, Quantum Inf. Processing, (2018), https://doi.org/10.1007/s11128-018-1898-6. Google Scholar

[18]

F. J. MacMilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[19]

R. C. Singleton, Maximum distance $q$-ary codes, IEEE Trans. Inform. Theory, 10 (1964), 116-118.  doi: 10.1109/tit.1964.1053661.  Google Scholar

[20]

B. Srinivasulu and M. Bhaintwal, $\mathbb{Z}_2(\mathbb{Z}_2+u\mathbb{Z}_2)$-Additive cyclic codes and their duals, Discrete Math. Algorithm. Appl., 8, (2016), 1650027, 19 pp. doi: 10.1142/S1793830916500270.  Google Scholar

[21]

Z.-X. Wan, Quaternary Codes, Series on Applied Mathematics, 8. World Scientific Publishing Co., Inc., River Edge, NJ, 1997. doi: 10.1142/9789812798121.  Google Scholar

[22]

S. X. ZhuY. Whang and M. J. Shi, Some results on cyclic codes over $\mathbb{F}_2+v\mathbb{F}_2$, IEEE Trans. Inform. Theory, 56 (2010), 1680-1684.  doi: 10.1109/TIT.2010.2040896.  Google Scholar

Table 1.  Some optimal $ \mathbb{Z}_p\mathbb{Z}_p[v] $-linear codes $ [n,k,d] $
$p$ $[\alpha,\beta]$ Generators $(\alpha+\beta,p^k,d_L)$ $[n,k,d]$
$3$ $[4,4]$ $f=x^3+2x^2+x+2,l=x^2+x+2,g_1=x+2,g_2=x+1$ $(8,3^7,4)$ $[12,7,4]$
$3$ $[4,4]$ $f=x^3+2x^2+x+2,l=x^2+x+2,g_1=x+2,g_2=1$ $(8,3^8,3)$ $[12,8,3]$
$5$ $[6,3]$ $f=x^5+4x^4+x^3+4x^2+x+4,l=x^4+3x^3+x+3,g_1=x^2+x+1,g_2=1$ $(9,5^5,6)$ $[12,5,6]$
$7$ $[2,6]$ $f=x^2+6,l=4x+6,g_1=x+5,g_2=x+4$ $(8,7^{10},4)$ $[14,10,4]$
$3$ $[5,5]$ $f=x^5+2,l=x^4+2x^3+x+2,g_1=x+2,g_2=1$ $(10,3^9,4)$ $[15,9,4]$
$5$ $[5,5]$ $f=x^5+4,l=x^4+2x^3+4x^2+x+3,g_1=x^2+3x+1,g_2=1$ $(10,5^8,6)$ $[15,8,6]$
$5$ $[6,12]$ $f=x^3+3x^2+2x+4,l=4x^2+3x+2,g_1=x+4,g_2=x+3$ $(18,5^{25},4)$ $[30,25,4]$
$p$ $[\alpha,\beta]$ Generators $(\alpha+\beta,p^k,d_L)$ $[n,k,d]$
$3$ $[4,4]$ $f=x^3+2x^2+x+2,l=x^2+x+2,g_1=x+2,g_2=x+1$ $(8,3^7,4)$ $[12,7,4]$
$3$ $[4,4]$ $f=x^3+2x^2+x+2,l=x^2+x+2,g_1=x+2,g_2=1$ $(8,3^8,3)$ $[12,8,3]$
$5$ $[6,3]$ $f=x^5+4x^4+x^3+4x^2+x+4,l=x^4+3x^3+x+3,g_1=x^2+x+1,g_2=1$ $(9,5^5,6)$ $[12,5,6]$
$7$ $[2,6]$ $f=x^2+6,l=4x+6,g_1=x+5,g_2=x+4$ $(8,7^{10},4)$ $[14,10,4]$
$3$ $[5,5]$ $f=x^5+2,l=x^4+2x^3+x+2,g_1=x+2,g_2=1$ $(10,3^9,4)$ $[15,9,4]$
$5$ $[5,5]$ $f=x^5+4,l=x^4+2x^3+4x^2+x+3,g_1=x^2+3x+1,g_2=1$ $(10,5^8,6)$ $[15,8,6]$
$5$ $[6,12]$ $f=x^3+3x^2+2x+4,l=4x^2+3x+2,g_1=x+4,g_2=x+3$ $(18,5^{25},4)$ $[30,25,4]$
Table 2.  Some MDSS codes $ (\alpha+\beta,p^k,d_L) $
$ p $ $ [\alpha,\beta] $ Generators $ (\alpha+\beta,p^k,d_L) $
$ 3 $ $ [3,3] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (6,3^8,2) $
$ 5 $ $ [4,4] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (8,5^{11},2) $
$ 11 $ $ [7,8] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (15,11^{22},2) $
$ 29 $ $ [12,6] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (18,29^{23},2) $
$ 37 $ $ [29,31] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (60,37^{90},2) $
$ 59 $ $ [36,68] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (104,59^{171},2) $
$ 97 $ $ [106,93] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (199,97^{291},2) $
$ p $ $ [\alpha,\beta] $ Generators $ (\alpha+\beta,p^k,d_L) $
$ 3 $ $ [3,3] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (6,3^8,2) $
$ 5 $ $ [4,4] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (8,5^{11},2) $
$ 11 $ $ [7,8] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (15,11^{22},2) $
$ 29 $ $ [12,6] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (18,29^{23},2) $
$ 37 $ $ [29,31] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (60,37^{90},2) $
$ 59 $ $ [36,68] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (104,59^{171},2) $
$ 97 $ $ [106,93] $ $ f=x-1,l=1,g_1=1,g_2=1 $ $ (199,97^{291},2) $
Table 3.  Quantum codes $ [[N,K,\geq D]]_p $}
$[\alpha, \beta]$ $f$ $g_1$ $g_2$ $(\alpha+\beta, p^k, d_L)$ $[[N, K, \geq D]]_p$ $[[N', K', D']]_p$
$[5, 5]$ 1, 3, 1 1, 3, 1 1, 3, 1 $(10, 5^9, 3)$ $[[15, 3, \geq3]]_5$ -
$[20, 5]$ 1, 3, 2, 3, 1 1, 3, 1 1, 3, 1 $(25, 5^{22}, 3)$ $[[30, 14, \geq3]]_5$ $[[10, 4, 3]]_5$ (ref.[12])
$[11, 11]$ 1, 7, 6, 7, 1 1, 7, 6, 7, 1 1, 7, 6, 7, 1 $(22, 11^{21}, 5)$ $[[33, 9, \geq5]]_{11}$ -
$[21, 7]$ 1, 6, 0, 6, 1 1, 5, 1 1, 5, 1 $(28, 7^{27}, 3)$ $[[35, 19, \geq3]]_7$ $[[18, 2, 3]]_7$ (ref.[16])
$[14, 14]$ 1, 1, 5, 5, 1, 1 1, 1, 5, 5, 1, 1 1, 1, 5, 5, 1, 1 $(28, 7^{27}, 4)$ $[[42, 12, \geq4]]_7$ $[[11, 1, 4]]_7$ (ref.[11])
$[9, 18]$ 1, 2, 0, 2, 1 1, 0, 2, 2, 0, 1 1, 0, 2, 2, 0, 1 $(27, 3^{31}, 3)$ $[[45, 17, \geq3]]_3$ -
$[17, 17]$ 1, 13, 6, 13, 1 1, 13, 6, 13, 1 1, 13, 6, 13, 1 $(34, 17^{39}, 5)$ $[[51, 27, \geq5]]_{17}$ $[[48, 24, 5]]_{17}$ (ref.[14])
$[26, 13]$ 1, 10, 2, 2, 10, 1 1, 9, 6, 9, 1 1, 9, 6, 9, 1 $(39, 13^{39}, 4)$ $[[52, 26, \geq4]]_{13}$ $[[24, 12, 4]]_{13}$ (ref.[14])
$[20, 20]$ 1, 3, 2, 3, 1 1, 3, 2, 3, 1 1, 3, 2, 3, 1 $(40, 5^{48}, 3)$ $[[60, 36, \geq3]]_5$ $[[60, 56, 2]]_5$ (ref.[12])
$[22, 22]$ 1, 1, 9, 9, 1, 1 1, 1, 9, 9, 1, 1 1, 1, 9, 9, 1, 1 $(44, 11^{51}, 4)$ $[[66, 36, \geq4]]_{11}$ $[[52, 28, 3]]_{11}$(ref.[16])
$[30, 30]$ 1, 2, 4, 2, 1 1, 2, 4, 2, 1 1, 2, 4, 2, 1 $(60, 5^{78}, 3)$ $[[90, 66, \geq3]]_5$ $[[30, 20, 3]]_5$ (ref.[17])
$[46, 23]$ 1, 22, 22, 1 1, 21, 1 1, 21, 1 $(69, 23^{85}, 3)$ $[[92, 78, \geq3]]_{23}$ $[[48, 40, \geq3]]_{23}$ (ref.[13])
$[31, 31]$ 1, 29, 1 1, 29, 1 1, 29, 1 $(62, 31^{87}, 3)$ $[[93, 81, \geq3]]_{31}$ $[[52, 44, \geq3]]_{31}$ (ref.[13])
$[47, 47]$ 1, 45, 1 1, 45, 1 1, 45, 1 $(94, 47^{135}, 3)$ $[[141,129, \geq3]]_{47}$ -
$[59, 59]$ 1, 57, 1 1, 57, 1 1, 57, 1 $(118, 59^{171}, 3)$ $[[177,165, \geq3]]_{59}$ -
$[\alpha, \beta]$ $f$ $g_1$ $g_2$ $(\alpha+\beta, p^k, d_L)$ $[[N, K, \geq D]]_p$ $[[N', K', D']]_p$
$[5, 5]$ 1, 3, 1 1, 3, 1 1, 3, 1 $(10, 5^9, 3)$ $[[15, 3, \geq3]]_5$ -
$[20, 5]$ 1, 3, 2, 3, 1 1, 3, 1 1, 3, 1 $(25, 5^{22}, 3)$ $[[30, 14, \geq3]]_5$ $[[10, 4, 3]]_5$ (ref.[12])
$[11, 11]$ 1, 7, 6, 7, 1 1, 7, 6, 7, 1 1, 7, 6, 7, 1 $(22, 11^{21}, 5)$ $[[33, 9, \geq5]]_{11}$ -
$[21, 7]$ 1, 6, 0, 6, 1 1, 5, 1 1, 5, 1 $(28, 7^{27}, 3)$ $[[35, 19, \geq3]]_7$ $[[18, 2, 3]]_7$ (ref.[16])
$[14, 14]$ 1, 1, 5, 5, 1, 1 1, 1, 5, 5, 1, 1 1, 1, 5, 5, 1, 1 $(28, 7^{27}, 4)$ $[[42, 12, \geq4]]_7$ $[[11, 1, 4]]_7$ (ref.[11])
$[9, 18]$ 1, 2, 0, 2, 1 1, 0, 2, 2, 0, 1 1, 0, 2, 2, 0, 1 $(27, 3^{31}, 3)$ $[[45, 17, \geq3]]_3$ -
$[17, 17]$ 1, 13, 6, 13, 1 1, 13, 6, 13, 1 1, 13, 6, 13, 1 $(34, 17^{39}, 5)$ $[[51, 27, \geq5]]_{17}$ $[[48, 24, 5]]_{17}$ (ref.[14])
$[26, 13]$ 1, 10, 2, 2, 10, 1 1, 9, 6, 9, 1 1, 9, 6, 9, 1 $(39, 13^{39}, 4)$ $[[52, 26, \geq4]]_{13}$ $[[24, 12, 4]]_{13}$ (ref.[14])
$[20, 20]$ 1, 3, 2, 3, 1 1, 3, 2, 3, 1 1, 3, 2, 3, 1 $(40, 5^{48}, 3)$ $[[60, 36, \geq3]]_5$ $[[60, 56, 2]]_5$ (ref.[12])
$[22, 22]$ 1, 1, 9, 9, 1, 1 1, 1, 9, 9, 1, 1 1, 1, 9, 9, 1, 1 $(44, 11^{51}, 4)$ $[[66, 36, \geq4]]_{11}$ $[[52, 28, 3]]_{11}$(ref.[16])
$[30, 30]$ 1, 2, 4, 2, 1 1, 2, 4, 2, 1 1, 2, 4, 2, 1 $(60, 5^{78}, 3)$ $[[90, 66, \geq3]]_5$ $[[30, 20, 3]]_5$ (ref.[17])
$[46, 23]$ 1, 22, 22, 1 1, 21, 1 1, 21, 1 $(69, 23^{85}, 3)$ $[[92, 78, \geq3]]_{23}$ $[[48, 40, \geq3]]_{23}$ (ref.[13])
$[31, 31]$ 1, 29, 1 1, 29, 1 1, 29, 1 $(62, 31^{87}, 3)$ $[[93, 81, \geq3]]_{31}$ $[[52, 44, \geq3]]_{31}$ (ref.[13])
$[47, 47]$ 1, 45, 1 1, 45, 1 1, 45, 1 $(94, 47^{135}, 3)$ $[[141,129, \geq3]]_{47}$ -
$[59, 59]$ 1, 57, 1 1, 57, 1 1, 57, 1 $(118, 59^{171}, 3)$ $[[177,165, \geq3]]_{59}$ -
[1]

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