American Institute of Mathematical Sciences

Advanced
doi: 10.3934/amc.2020134

Character sums over a non-chain ring and their applications

Liqin Qian and  Xiwang Cao ,

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 211106, China

* Corresponding author: Xiwang Cao

Received  June 2020 Revised  November 2020 Published  March 2021

Fund Project: This work is supported by the National Natural Science Foundation of China under Grant 11771007

Some valuable results over rings have a promising utilization in coding theory and error-correcting code theory. In this paper, we study character sums over a certain non-chain ring and their applications in codebooks. There are two major ingredients in this study. The first ingredient is to investigate Gaussian sums, hyper Eisenstein sums, Jacobi sums over a certain non-chain ring and study the properties of these character sums. For their applications, the second ingredient is to present three classes of asymptotically optimal codebooks with respect to the Welch bound and a family of optimal codebooks with respect to the Levenshtein bound, which are constructed from character sums over a certain non-chain ring.

Keywords: Non-chain ring, character sum, codebook, Welch bound, Levenshtein bound.
Mathematics Subject Classification: Primary: 11T24, 11L40, 11T71; Secondary: 11L05.
Citation: Liqin Qian, Xiwang Cao. Character sums over a non-chain ring and their applications. Advances in Mathematics of Communications, doi: 10.3934/amc.2020134
References:
[1]

A. R. Calderbank, P. J. Cameron, W. M. Kantor and J. J. Seidel, $\mathbb{Z}_4$-Kerdock codes, orthogonal spreads, and extremal Euclidean linesets, Proc. London Math. Soc., 75 (1997), 436-480.  doi: 10.1112/S0024611597000403.  Google Scholar

[2]

X. Cao, W. Chou and X. Zhang, More constructions of near optimal codebooks associated with binary sequences, Adv. Math. Commun., 11 (2017), 187-202.  doi: 10.3934/amc.2017012.  Google Scholar

[3]

E. Candes and M. Wakin, An introduction to compressive sampling, IEEE Signal Process, 25 (2008), 21-30.   Google Scholar

[4]

C. Ding, Complex codebooks from combinatorial designs, IEEE Trans, Inform. Theory, 52 (2006), 4229-4235.  doi: 10.1109/TIT.2006.880058.  Google Scholar

[5]

C. Ding, The weight distribution of some irreducible cyclic codes, IEEE Trans. Inform. Theory, 55 (2009), 955-960.  doi: 10.1109/TIT.2008.2011511.  Google Scholar

[6]

C. Ding and T. Feng, A generic construction of complex codebooks meeting the Welch bound, IEEE Trans. Inform. Theory, 53 (2007), 4245-4250.  doi: 10.1109/TIT.2007.907343.  Google Scholar

[7]

C. Ding and T. Feng, Codebooks from almost difference sets, Des. Codes Crypt., 46 (2008), 113-126.  doi: 10.1007/s10623-007-9140-z.  Google Scholar

[8]

C. Ding, Y. Gao and Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inform. Theory, 59 (2013), 7940-7946.  doi: 10.1109/TIT.2013.2281205.  Google Scholar

[9]

C. Ding and J. Yin, Signal sets from functions with optimum nonlinearity, IEEE Trans. Commun., 55 (2007), 936-940.  doi: 10.1109/TCOMM.2007.894113.  Google Scholar

[10]

H. Hu and J. Wu, New constructions of codebooks nearly meeting the Welch bound with equality, IEEE Trans. Inform. Theory, 60 (2014), 1348-1355.  doi: 10.1109/TIT.2013.2292745.  Google Scholar

[11]

S. Hong, H. Park, T. Helleseth and Y. Kim, Near optimal partial Hadamard codebook construction using binary sequences obtained from quadratic residue mapping, IEEE Trans. Inform. Theory, 60 (2014), 3698-3705.  doi: 10.1109/TIT.2014.2314298.  Google Scholar

[12]

Z. Heng, C. Ding and Q. Yue, New constructions of asymptotically optimal codebooks with multiplicative characters, IEEE Trans. Inform. Theory, 63 (2017), 6179-6187.  doi: 10.1109/TIT.2017.2693204.  Google Scholar

[13]

Z. Heng and Q. Yue, Optimal codebooks achieving the Levenshtein bound from generalized bent functions over $\mathbb{Z}_4$, Cryptogr. Commun., 9 (2017), 41-53.  doi: 10.1007/s12095-016-0194-5.  Google Scholar

[14]

Z. Heng, Nearly optimal codebooks based on generalized Jacobi sums, Discrete Appied Mathematics, 250 (2018), 227-240.  doi: 10.1016/j.dam.2018.05.017.  Google Scholar

[15]

J. Kovacevic and A. Chebira, An introduction to frames, Found. Trends Signal Process., 2 (2008), 1-94.   Google Scholar

[16]

G. Luo and X. Cao, Two constructions of asymptotically optimal codebooks, Crypt. Commun., 11 (2019), 825-838.  doi: 10.1007/s12095-018-0331-4.  Google Scholar

[17]

G. Luo and X. Cao, Two constructions of asymptotically optimal codebooks via the hyper Eisenstein sum, IEEE Trans. Inform. Theory, 64 (2018), 6498-6505.  doi: 10.1109/TIT.2017.2777492.  Google Scholar

[18]

G. Luo and X. Cao, New constructions of codebooks asymptotically achieving the Welch bound, 2018 IEEE International Symposium on Information Theory (ISIT)-Vail, (2018), 2346-2350. doi: 10.1109/ISIT.2018.8437838.  Google Scholar

[19]

Y. Liu, M. Shi and P. Solé, Two-weight and three weight codes from trace codes over $\mathbb{F}_p+u\mathbb{F}_p+v\mathbb{F}_p+uv\mathbb{F}_p$, Discrete Mathematics, 341 (2018), 350-357.  doi: 10.1016/j.disc.2017.09.003.  Google Scholar

[20]

W. Lu, X. Wu, X. Cao and M. Chen, Six constructions of asymptotically optimal codebooks via the character sums, Des. Codes Cryptogr., 88 (2020), 1139-1158.  doi: 10.1007/s10623-020-00735-w.  Google Scholar

[21]

C. Li, Q. Yue and Y. Huang, Two families of nearly optimal codebooks, Des. Codes Cryptogr., 75 (2015), 43-57.  doi: 10.1007/s10623-013-9891-7.  Google Scholar

[22] R. Lidl, H. Niederreiter and P. M. Cohn, Finite Fields, Cambridge University Press, 1997.   Google Scholar
[23]

J. Li, S. Zhu and K. Feng, The Gauss sums and Jacobi sums over Galois ring $GR(p^2, r)$, Science China Mathematics, 56 (2013), 1457-1465.  doi: 10.1007/s11425-013-4629-6.  Google Scholar

[24]

S. Mesnager, Several new infinite families of bent functions and their duals, IEEE Trans. Inform. Theory, 60 (2014), 4397-4407.  doi: 10.1109/TIT.2014.2320974.  Google Scholar

[25]

S. Mesnager and A. Sinak, Several classes of minimal linear codes with few weights from weakly regular plateaued function, IEEE Trans. Inform. Theory, 66 (2020), 2296-2310.  doi: 10.1109/TIT.2019.2956130.  Google Scholar

[26]

J. Massey and T. Mittelholzer, Welch bound and sequence sets for code-division multiple-access systems, Sequences II, (1999), 63-78.  Google Scholar

[27]

J. Renes, R. Blume-Kohout, A. Scot and C. Caves, Symmetric informationally complete quantum measurements, J. Math. Phys., 45 (2004), 2171-2180. doi: 10.1063/1.1737053.  Google Scholar

[28]

D. Sarwate, Meeting the Welch bound with equality, in Sequences and their Applications, New York, Springer-Verlag, 1999, 79-102. Google Scholar

[29]

T. Strohmer and R. W. Heath, Grassmannian frames with applications to coding and communication, Appl. Comput. Harmon. Anal., 14 (2003), 257-275.  doi: 10.1016/S1063-5203(03)00023-X.  Google Scholar

[30]

M. Shi, R. Wu, L. Qian, L. Sok and P. Solé, New classes of $p$-Ary few weight codes, Bulletin of the Malaysian Mathematical Sciences Society, 42 (2019), 1393-1412.  doi: 10.1007/s40840-017-0553-1.  Google Scholar

[31]

M. Shi, L. Qian and P. Solé, New few weight codes from trace codes over a local Ring, Applicable Algebra in Engineering, Communication and Computing, 29 (2018), 335-350.  doi: 10.1007/s00200-017-0345-8.  Google Scholar

[32]

P. Tan, Z. Zhou and D. Zhang, A construction of codebooks nearly achieving the Levenshtein bound, IEEE Signal Process. Lett., 23 (2016), 1306-1309.   Google Scholar

[33]

L. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Theory, 20 (1974), 397-399.  doi: 10.1109/TIT.1974.1055219.  Google Scholar

[34]

W. K. Wootters and B. D. Fields, Optimal state-determination by mutually unbiased measurements, Ann. Phys., 191 (1989), 363-381.  doi: 10.1016/0003-4916(89)90322-9.  Google Scholar

[35]

C. Xiang, C. Ding and S. Mesnager, Optimal codebooks from binary codes meeting the Levenshtein bound, IEEE Trans. Inform. Theory, 61 (2015), 6526-6535.  doi: 10.1109/TIT.2015.2487451.  Google Scholar

[36]

N. Yu, A construction of codebooks associated with binary sequences, IEEE Trans. Inform. Theory, 58 (2012), 5522-5533.  doi: 10.1109/TIT.2012.2196021.  Google Scholar

[37]

Z. Zhou, C. Ding and N. Li, New families of codebooks achieving the Levenstein bound, IEEE Trans. Inform. Theory, 60 (2014), 7382-7387.  doi: 10.1109/TIT.2014.2353052.  Google Scholar

[38]

A. Zhang and K. Feng, Two classes of codebooks nearly meeting the Welch bound, IEEE Trans. Inform. Theory, 58 (2012), 2507-2511.  doi: 10.1109/TIT.2011.2176531.  Google Scholar

[39]

A. Zhang and K. Feng, Construction of cyclotomic codebooks nearly meeting the Welch bound, Des. Codes Cryptogr., 63 (2013), 209-224.  doi: 10.1007/s10623-011-9549-2.  Google Scholar

[40]

Z. Zhou and X. Tang, New nearly optimal codebooks from relative difference sets, Adv. Math. Commun., 5 (2011), 521-527.  doi: 10.3934/amc.2011.5.521.  Google Scholar

show all references

References:
[1]

A. R. Calderbank, P. J. Cameron, W. M. Kantor and J. J. Seidel, $\mathbb{Z}_4$-Kerdock codes, orthogonal spreads, and extremal Euclidean linesets, Proc. London Math. Soc., 75 (1997), 436-480.  doi: 10.1112/S0024611597000403.  Google Scholar

[2]

X. Cao, W. Chou and X. Zhang, More constructions of near optimal codebooks associated with binary sequences, Adv. Math. Commun., 11 (2017), 187-202.  doi: 10.3934/amc.2017012.  Google Scholar

[3]

E. Candes and M. Wakin, An introduction to compressive sampling, IEEE Signal Process, 25 (2008), 21-30.   Google Scholar

[4]

C. Ding, Complex codebooks from combinatorial designs, IEEE Trans, Inform. Theory, 52 (2006), 4229-4235.  doi: 10.1109/TIT.2006.880058.  Google Scholar

[5]

C. Ding, The weight distribution of some irreducible cyclic codes, IEEE Trans. Inform. Theory, 55 (2009), 955-960.  doi: 10.1109/TIT.2008.2011511.  Google Scholar

[6]

C. Ding and T. Feng, A generic construction of complex codebooks meeting the Welch bound, IEEE Trans. Inform. Theory, 53 (2007), 4245-4250.  doi: 10.1109/TIT.2007.907343.  Google Scholar

[7]

C. Ding and T. Feng, Codebooks from almost difference sets, Des. Codes Crypt., 46 (2008), 113-126.  doi: 10.1007/s10623-007-9140-z.  Google Scholar

[8]

C. Ding, Y. Gao and Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inform. Theory, 59 (2013), 7940-7946.  doi: 10.1109/TIT.2013.2281205.  Google Scholar

[9]

C. Ding and J. Yin, Signal sets from functions with optimum nonlinearity, IEEE Trans. Commun., 55 (2007), 936-940.  doi: 10.1109/TCOMM.2007.894113.  Google Scholar

[10]

H. Hu and J. Wu, New constructions of codebooks nearly meeting the Welch bound with equality, IEEE Trans. Inform. Theory, 60 (2014), 1348-1355.  doi: 10.1109/TIT.2013.2292745.  Google Scholar

[11]

S. Hong, H. Park, T. Helleseth and Y. Kim, Near optimal partial Hadamard codebook construction using binary sequences obtained from quadratic residue mapping, IEEE Trans. Inform. Theory, 60 (2014), 3698-3705.  doi: 10.1109/TIT.2014.2314298.  Google Scholar

[12]

Z. Heng, C. Ding and Q. Yue, New constructions of asymptotically optimal codebooks with multiplicative characters, IEEE Trans. Inform. Theory, 63 (2017), 6179-6187.  doi: 10.1109/TIT.2017.2693204.  Google Scholar

[13]

Z. Heng and Q. Yue, Optimal codebooks achieving the Levenshtein bound from generalized bent functions over $\mathbb{Z}_4$, Cryptogr. Commun., 9 (2017), 41-53.  doi: 10.1007/s12095-016-0194-5.  Google Scholar

[14]

Z. Heng, Nearly optimal codebooks based on generalized Jacobi sums, Discrete Appied Mathematics, 250 (2018), 227-240.  doi: 10.1016/j.dam.2018.05.017.  Google Scholar

[15]

J. Kovacevic and A. Chebira, An introduction to frames, Found. Trends Signal Process., 2 (2008), 1-94.   Google Scholar

[16]

G. Luo and X. Cao, Two constructions of asymptotically optimal codebooks, Crypt. Commun., 11 (2019), 825-838.  doi: 10.1007/s12095-018-0331-4.  Google Scholar

[17]

G. Luo and X. Cao, Two constructions of asymptotically optimal codebooks via the hyper Eisenstein sum, IEEE Trans. Inform. Theory, 64 (2018), 6498-6505.  doi: 10.1109/TIT.2017.2777492.  Google Scholar

[18]

G. Luo and X. Cao, New constructions of codebooks asymptotically achieving the Welch bound, 2018 IEEE International Symposium on Information Theory (ISIT)-Vail, (2018), 2346-2350. doi: 10.1109/ISIT.2018.8437838.  Google Scholar

[19]

Y. Liu, M. Shi and P. Solé, Two-weight and three weight codes from trace codes over $\mathbb{F}_p+u\mathbb{F}_p+v\mathbb{F}_p+uv\mathbb{F}_p$, Discrete Mathematics, 341 (2018), 350-357.  doi: 10.1016/j.disc.2017.09.003.  Google Scholar

[20]

W. Lu, X. Wu, X. Cao and M. Chen, Six constructions of asymptotically optimal codebooks via the character sums, Des. Codes Cryptogr., 88 (2020), 1139-1158.  doi: 10.1007/s10623-020-00735-w.  Google Scholar

[21]

C. Li, Q. Yue and Y. Huang, Two families of nearly optimal codebooks, Des. Codes Cryptogr., 75 (2015), 43-57.  doi: 10.1007/s10623-013-9891-7.  Google Scholar

[22] R. Lidl, H. Niederreiter and P. M. Cohn, Finite Fields, Cambridge University Press, 1997.   Google Scholar
[23]

J. Li, S. Zhu and K. Feng, The Gauss sums and Jacobi sums over Galois ring $GR(p^2, r)$, Science China Mathematics, 56 (2013), 1457-1465.  doi: 10.1007/s11425-013-4629-6.  Google Scholar

[24]

S. Mesnager, Several new infinite families of bent functions and their duals, IEEE Trans. Inform. Theory, 60 (2014), 4397-4407.  doi: 10.1109/TIT.2014.2320974.  Google Scholar

[25]

S. Mesnager and A. Sinak, Several classes of minimal linear codes with few weights from weakly regular plateaued function, IEEE Trans. Inform. Theory, 66 (2020), 2296-2310.  doi: 10.1109/TIT.2019.2956130.  Google Scholar

[26]

J. Massey and T. Mittelholzer, Welch bound and sequence sets for code-division multiple-access systems, Sequences II, (1999), 63-78.  Google Scholar

[27]

J. Renes, R. Blume-Kohout, A. Scot and C. Caves, Symmetric informationally complete quantum measurements, J. Math. Phys., 45 (2004), 2171-2180. doi: 10.1063/1.1737053.  Google Scholar

[28]

D. Sarwate, Meeting the Welch bound with equality, in Sequences and their Applications, New York, Springer-Verlag, 1999, 79-102. Google Scholar

[29]

T. Strohmer and R. W. Heath, Grassmannian frames with applications to coding and communication, Appl. Comput. Harmon. Anal., 14 (2003), 257-275.  doi: 10.1016/S1063-5203(03)00023-X.  Google Scholar

[30]

M. Shi, R. Wu, L. Qian, L. Sok and P. Solé, New classes of $p$-Ary few weight codes, Bulletin of the Malaysian Mathematical Sciences Society, 42 (2019), 1393-1412.  doi: 10.1007/s40840-017-0553-1.  Google Scholar

[31]

M. Shi, L. Qian and P. Solé, New few weight codes from trace codes over a local Ring, Applicable Algebra in Engineering, Communication and Computing, 29 (2018), 335-350.  doi: 10.1007/s00200-017-0345-8.  Google Scholar

[32]

P. Tan, Z. Zhou and D. Zhang, A construction of codebooks nearly achieving the Levenshtein bound, IEEE Signal Process. Lett., 23 (2016), 1306-1309.   Google Scholar

[33]

L. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Theory, 20 (1974), 397-399.  doi: 10.1109/TIT.1974.1055219.  Google Scholar

[34]

W. K. Wootters and B. D. Fields, Optimal state-determination by mutually unbiased measurements, Ann. Phys., 191 (1989), 363-381.  doi: 10.1016/0003-4916(89)90322-9.  Google Scholar

[35]

C. Xiang, C. Ding and S. Mesnager, Optimal codebooks from binary codes meeting the Levenshtein bound, IEEE Trans. Inform. Theory, 61 (2015), 6526-6535.  doi: 10.1109/TIT.2015.2487451.  Google Scholar

[36]

N. Yu, A construction of codebooks associated with binary sequences, IEEE Trans. Inform. Theory, 58 (2012), 5522-5533.  doi: 10.1109/TIT.2012.2196021.  Google Scholar

[37]

Z. Zhou, C. Ding and N. Li, New families of codebooks achieving the Levenstein bound, IEEE Trans. Inform. Theory, 60 (2014), 7382-7387.  doi: 10.1109/TIT.2014.2353052.  Google Scholar

[38]

A. Zhang and K. Feng, Two classes of codebooks nearly meeting the Welch bound, IEEE Trans. Inform. Theory, 58 (2012), 2507-2511.  doi: 10.1109/TIT.2011.2176531.  Google Scholar

[39]

A. Zhang and K. Feng, Construction of cyclotomic codebooks nearly meeting the Welch bound, Des. Codes Cryptogr., 63 (2013), 209-224.  doi: 10.1007/s10623-011-9549-2.  Google Scholar

[40]

Z. Zhou and X. Tang, New nearly optimal codebooks from relative difference sets, Adv. Math. Commun., 5 (2011), 521-527.  doi: 10.3934/amc.2011.5.521.  Google Scholar

Table 1.  Parameters of the $ (N_1, K_1) $ codebook of Theorem 5.1
$ q $ $ (N_1, K_1) $ $ I_{\max}(C_1(R^*, \widehat{R}^*\times \widehat{R})) $ $ I_W $ $ \frac{I_{\max}(C_1(R^*, \widehat{R}^*\times \widehat{R}))}{I_W} $
$ 13 $ $ (2028,144) $ 0.0902778 0.0803401 1.123695
$ 23 $ $ (11638,484) $ 0.0475207 0.0445012 1.067852
$ 5^2 $ $ (15000,576) $ 0.0434028 0.0408602 1.062227
$ 7^2 $ $ (115248,2304) $ 0.0212674 0.0206241 1.031192
$ 3^4 $ $ (524880,6400) $ 0.0126563 0.0124236 1.018730
$ 103 $ $ (1082118,10404) $ 0.00990004 0.00975668 1.014694
$ 151 $ $ (3420150,22500) $ 0.00666667 0.00664470 1.003311
$ 5^4 $ $ (243750000,389376) $ 0.00160513 0.00160128 1.002404
$ 7^5 $ $ (4.747279e+12,282441636) $ 0.0000595061 0.0000595008 1.000089
Table Options

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$ q $ $ (N_1, K_1) $ $ I_{\max}(C_1(R^*, \widehat{R}^*\times \widehat{R})) $ $ I_W $ $ \frac{I_{\max}(C_1(R^*, \widehat{R}^*\times \widehat{R}))}{I_W} $
$ 13 $ $ (2028,144) $ 0.0902778 0.0803401 1.123695
$ 23 $ $ (11638,484) $ 0.0475207 0.0445012 1.067852
$ 5^2 $ $ (15000,576) $ 0.0434028 0.0408602 1.062227
$ 7^2 $ $ (115248,2304) $ 0.0212674 0.0206241 1.031192
$ 3^4 $ $ (524880,6400) $ 0.0126563 0.0124236 1.018730
$ 103 $ $ (1082118,10404) $ 0.00990004 0.00975668 1.014694
$ 151 $ $ (3420150,22500) $ 0.00666667 0.00664470 1.003311
$ 5^4 $ $ (243750000,389376) $ 0.00160513 0.00160128 1.002404
$ 7^5 $ $ (4.747279e+12,282441636) $ 0.0000595061 0.0000595008 1.000089
Table 2.  Parameters of the $ (N_2, K_2) $ codebook of Theorem 5.2
$ q $ $ (N_2, K_2) $ $ I_{\max}(C_2(R^*, \widehat{R}^*\times \widehat{R})) $ $ I_W $ $ \frac{I_{\max}(C_2(R^*, \widehat{R}^*\times \widehat{R}))}{I_W} $
$ 17 $ $ (4352,256) $ 0.0664062 0.0606409 1.095074
$ 37 $ $ (47952,1296) $ 0.0285494 0.0274001 1.041944
$ 83 $ $ (558092,6724) $ 0.0123438 0.0121214 1.018347
$ 11^2 $ $ (1742400,14400) $ 0.00840278 0.00829883 1.012526
$ 3^5 $ $ (14231052,58564) $ 0.00414931 0.00412372 1.006205
$ 293 $ $ (24982352,85264) $ 0.00343639 0.00341881 1.005141
$ 7^3 $ $ (40118652,116964) $ 0.00293253 0.00291971 1.004389
$ 13^3 $ $ (1.05948e+10,4822416) $ 0.000455581 0.00045527 1.000683
$ 5^5 $ $ (3.0498e+10,9759376) $ 0.000320205 0.000320051 1.000480
Table Options

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$ q $ $ (N_2, K_2) $ $ I_{\max}(C_2(R^*, \widehat{R}^*\times \widehat{R})) $ $ I_W $ $ \frac{I_{\max}(C_2(R^*, \widehat{R}^*\times \widehat{R}))}{I_W} $
$ 17 $ $ (4352,256) $ 0.0664062 0.0606409 1.095074
$ 37 $ $ (47952,1296) $ 0.0285494 0.0274001 1.041944
$ 83 $ $ (558092,6724) $ 0.0123438 0.0121214 1.018347
$ 11^2 $ $ (1742400,14400) $ 0.00840278 0.00829883 1.012526
$ 3^5 $ $ (14231052,58564) $ 0.00414931 0.00412372 1.006205
$ 293 $ $ (24982352,85264) $ 0.00343639 0.00341881 1.005141
$ 7^3 $ $ (40118652,116964) $ 0.00293253 0.00291971 1.004389
$ 13^3 $ $ (1.05948e+10,4822416) $ 0.000455581 0.00045527 1.000683
$ 5^5 $ $ (3.0498e+10,9759376) $ 0.000320205 0.000320051 1.000480
Table 3.  Parameters of the $ (N_3, K_3) $ codebook of Theorem 5.3
$ q $ $ (N_3, K_3) $ $ I_{\max}({C}_3(D', \widehat{R}^*\times\widehat{R}^*)) $ $ I_W $ $ \frac{I_{\max}({C}_3(D', \widehat{R}^*\times\widehat{R}^*))}{I_W} $
$ 19 $ $ (5832,324) $ 0.0657439 0.0573525 1.146314
$ 59 $ $ (195112,3364) $ 0.0181594 0.0173972 1.043812
$ 3^4 $ $ (512000,6400) $ 0.0129787 0.0125809 1.031622
$ 113 $ $ (1404928,12544) $ 0.00917133 0.00896942 1.022511
$ 211 $ $ (9261000,44100) $ 0.00483048 0.00477339 1.011959
$ 281 $ $ (21952000,78400) $ 0.00360992 0.00357787 1.008959
$ 5^4 $ $ (242970624,389376) $ 0.00161029 0.00160385 1.004013
$ 11^3 $ $ (2.35264e+9,1768900) $ 0.000753578 0.000752163 1.001881
$ 17^3 $ $ (1.18515e+11,24127744) $ 0.000203707 0.000203604 1.000509
Table Options

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$ q $ $ (N_3, K_3) $ $ I_{\max}({C}_3(D', \widehat{R}^*\times\widehat{R}^*)) $ $ I_W $ $ \frac{I_{\max}({C}_3(D', \widehat{R}^*\times\widehat{R}^*))}{I_W} $
$ 19 $ $ (5832,324) $ 0.0657439 0.0573525 1.146314
$ 59 $ $ (195112,3364) $ 0.0181594 0.0173972 1.043812
$ 3^4 $ $ (512000,6400) $ 0.0129787 0.0125809 1.031622
$ 113 $ $ (1404928,12544) $ 0.00917133 0.00896942 1.022511
$ 211 $ $ (9261000,44100) $ 0.00483048 0.00477339 1.011959
$ 281 $ $ (21952000,78400) $ 0.00360992 0.00357787 1.008959
$ 5^4 $ $ (242970624,389376) $ 0.00161029 0.00160385 1.004013
$ 11^3 $ $ (2.35264e+9,1768900) $ 0.000753578 0.000752163 1.001881
$ 17^3 $ $ (1.18515e+11,24127744) $ 0.000203707 0.000203604 1.000509
[1]

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