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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 |
$ \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.
References:
[1] |
T. Abualrub, I. 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. |
[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. |
[4] |
I. Aydogdu, T. 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. |
[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. |
[6] |
I. Aydogdu, T. 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. |
[7] |
J. Borges, C. Fernández-Córdoba, J. 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. |
[8] |
J. Borges, C. 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. |
[9] |
A. R. Calderbank, E. M. Rains, P. 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. |
[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. |
[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. |
[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. |
[14] |
Y. Gao, J. 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. |
[15] |
A. R. Hammons, P. Kumar, A. R. Calderbank, N. 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. |
[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.
|
[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. |
[19] |
R. C. Singleton,
Maximum distance $q$-ary codes, IEEE Trans. Inform. Theory, 10 (1964), 116-118.
doi: 10.1109/tit.1964.1053661. |
[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. |
[21] |
Z.-X. Wan, Quaternary Codes, Series on Applied Mathematics, 8. World Scientific Publishing Co., Inc., River Edge, NJ, 1997.
doi: 10.1142/9789812798121. |
[22] |
S. X. Zhu, Y. 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. |
show all references
References:
[1] |
T. Abualrub, I. 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. |
[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. |
[4] |
I. Aydogdu, T. 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. |
[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. |
[6] |
I. Aydogdu, T. 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. |
[7] |
J. Borges, C. Fernández-Córdoba, J. 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. |
[8] |
J. Borges, C. 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. |
[9] |
A. R. Calderbank, E. M. Rains, P. 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. |
[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. |
[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. |
[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. |
[14] |
Y. Gao, J. 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. |
[15] |
A. R. Hammons, P. Kumar, A. R. Calderbank, N. 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. |
[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.
|
[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. |
[19] |
R. C. Singleton,
Maximum distance $q$-ary codes, IEEE Trans. Inform. Theory, 10 (1964), 116-118.
doi: 10.1109/tit.1964.1053661. |
[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. |
[21] |
Z.-X. Wan, Quaternary Codes, Series on Applied Mathematics, 8. World Scientific Publishing Co., Inc., River Edge, NJ, 1997.
doi: 10.1142/9789812798121. |
[22] |
S. X. Zhu, Y. 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. |
$p$ | $[\alpha,\beta]$ | 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 | $(\alpha+\beta,p^k,d_L)$ | $[n,k,d]$ |
$3$ | $[4,4]$ | $f=x^3+2x^2+x+2,l=x^2+x+2,g_1=x+2,g_2=x+1$ | $(8,3^7,4)$ | $[12,7,4]$ |
$3$ | $[4,4]$ | $f=x^3+2x^2+x+2,l=x^2+x+2,g_1=x+2,g_2=1$ | $(8,3^8,3)$ | $[12,8,3]$ |
$5$ | $[6,3]$ | $f=x^5+4x^4+x^3+4x^2+x+4,l=x^4+3x^3+x+3,g_1=x^2+x+1,g_2=1$ | $(9,5^5,6)$ | $[12,5,6]$ |
$7$ | $[2,6]$ | $f=x^2+6,l=4x+6,g_1=x+5,g_2=x+4$ | $(8,7^{10},4)$ | $[14,10,4]$ |
$3$ | $[5,5]$ | $f=x^5+2,l=x^4+2x^3+x+2,g_1=x+2,g_2=1$ | $(10,3^9,4)$ | $[15,9,4]$ |
$5$ | $[5,5]$ | $f=x^5+4,l=x^4+2x^3+4x^2+x+3,g_1=x^2+3x+1,g_2=1$ | $(10,5^8,6)$ | $[15,8,6]$ |
$5$ | $[6,12]$ | $f=x^3+3x^2+2x+4,l=4x^2+3x+2,g_1=x+4,g_2=x+3$ | $(18,5^{25},4)$ | $[30,25,4]$ |
Generators | |||
Generators | |||
$[\alpha, \beta]$ | $f$ | $g_1$ | $g_2$ | $(\alpha+\beta, p^k, d_L)$ | $[[N, K, \geq D]]_p$ | $[[N', K', D']]_p$ |
$[5, 5]$ | 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 | $(118, 59^{171}, 3)$ | $[[177,165, \geq3]]_{59}$ | - |
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