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doi: 10.3934/amc.2021059
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Generic constructions of MDS Euclidean self-dual codes via GRS codes

Ziteng Huang 1, , Weijun Fang 2,, , Fang-Wei Fu 3, and  Fengting Li 4,

1. 

Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
2. 

School of Cyber Science and Technology, Shandong University, Qingdao 266237, China
3. 

Chern Institute of Mathematics and LPMC, and Tianjin Key Laboratory of Network, and Data Security Technology, Nankai University, Tianjin 300071, China
4. 

Ant Group, Beijing 100020, China

* Corresponding author: Weijun Fang

Received  April 2021 Revised  September 2021 Early access December 2021

Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square $ q $, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of $ q $-ary MDS Euclidean self-dual codes of lengths in the form $ s\frac{q-1}{a}+t\frac{q-1}{b} $, where $ s $ and $ t $ range in some interval and $ a, b \,|\, (q -1) $. In particular, for large square $ q $, our constructions take up a proportion of generally more than 34% in all the possible lengths of $ q $-ary MDS Euclidean self-dual codes, which is larger than the previous results. Moreover, two new families of MDS Euclidean self-orthogonal codes and two new families of MDS Euclidean almost self-dual codes are given similarly.

Keywords: MDS codes, Euclidean self-dual codes, generalized Reed-Solomon codes, Euclidean self-orthogonal codes, Euclidean almost self-dual codes.
Mathematics Subject Classification: Primary: 94B05; Secondary: 94B60.
Citation: Ziteng Huang, Weijun Fang, Fang-Wei Fu, Fengting Li. Generic constructions of MDS Euclidean self-dual codes via GRS codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021059
References:
[1]

S. Balaji, M. Krishnan, M. Vajha, V. Ramkumar, B. Sasidharan and P. Kumar, Erasure coding for distributed storage: An overview, Sci. China Inf. Sci., 61 (2018), 100301.  doi: 10.1007/s11432-018-9482-6.  Google Scholar

[2]

J. H. Conway and N. J. A. Sloane, Sphere Packing, Lattices and Groups, 3$^rd$ edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-6568-7.  Google Scholar

[3]

R. Cramer, V. Daza, I. Gracia, J. J. Urroz, G. Leander, J. Marti-Farre and C. Padro, On codes, matroids and secure multiparty computation from linear secret sharing schemes, IEEE Trans. Inform. Theory, 54 (2008), 2644-2657.  doi: 10.1109/TIT.2008.921692.  Google Scholar

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S. T. Dougherty, S. Mesnager and P. Sole, Secret-sharing schemes based on self-dual codes, IEEE Information Theory Workshop, (2008), 338–342. doi: 10.1109/ITW.2008.4578681.  Google Scholar

[5]

Z. Du, C. Li and S. Mesnager, Constructions of self-orthogonal codes from hulls of BCH codes and their parameters, IEEE Trans. Inform. Theory, 66 (2020), 6774-6785.  doi: 10.1109/TIT.2020.2991635.  Google Scholar

[6]

W. Fang and F.-W. Fu, New constructions of MDS Euclidean self-dual codes from GRS codes and extended GRS codes, IEEE Trans. Inform. Theory, 65 (2019), 5574-5579.  doi: 10.1109/TIT.2019.2916367.  Google Scholar

[7]

W. Fang, F.-W. Fu, L. Li and S. Zhu, Euclidean and Hermitian hulls of MDS codes and their applications to EAQECCs, IEEE Trans. Inform. Theory, 66 (2020), 3527-3537.  doi: 10.1109/TIT.2019.2950245.  Google Scholar

[8]

X. Fang, K. Lebed, H. Liu and J. Luo, New MDS self-dual codes over finite fields of odd characteristic, Des. Codes Cryptogr., 88 (2020), 1127-1138.  doi: 10.1007/s10623-020-00734-x.  Google Scholar

[9]

X. Fang, M. Liu and J. Luo, New MDS Euclidean self-orthogonal codes, IEEE Trans. Inform. Theory, 67 (2021), 130-137.  doi: 10.1109/TIT.2020.3020986.  Google Scholar

[10]

W. Fang, S.-T. Xia and F.-W. Fu, Construction of MDS Euclidean self-dual codes via two subsets, IEEE Trans. Inform. Theory, 67 (2021), 5005-5015.  doi: 10.1109/TIT.2021.3085768.  Google Scholar

[11]

W. Fang, J. Zhang, S.-T. Xia and F.-W. Fu, A note on self-dual generalized Reed-Solomon codes, preprint, arXiv: 2005.11732 [cs.IT]. Google Scholar

[12]

M. Grassl and T. A. Gulliver, On self-dual MDS codes, IEEE Int. Symp. Inform. Theory, (2008), 1954–1957. doi: 10.1109/ISIT.2008.4595330.  Google Scholar

[13]

M. Harada and H. Kharaghani, Orthogonal designs, self-dual codes, and the Leech lattice, J. Combin. Des., 13 (2005), 184-194.  doi: 10.1002/jcd.20046.  Google Scholar

[14]

M. Harada and H. Kharaghani, Orthogonal designs and MDS self-dual codes, Australas. J. Combin., 35 (2006), 57-67.   Google Scholar

[15]

A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays: Theory and Applications, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1478-6.  Google Scholar

[16] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[17]

L. Jin and C. 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

[18]

K. Lebed, H. Liu and J. Luo, Construction of MDS self-dual codes over finite fields, Finite Fields Appl., 59 (2019), 199-207.  doi: 10.1016/j.ffa.2019.05.007.  Google Scholar

[19]

R. Li, Z. Xu and X. Zhao, On the classification of binary optimal self-orthogonal codes, IEEE Trans. Inform. Theory, 54 (2008), 3778-3782.  doi: 10.1109/TIT.2008.926367.  Google Scholar

[20]

G. Luo, X. Cao and X. Chen, MDS codes with hulls of arbitrary dimensions and their quantum error correction, IEEE Trans. Inform. Theory, 65 (2019), 2944-2952.  doi: 10.1109/TIT.2018.2874953.  Google Scholar

[21]

H. Tong and X. Wang, New MDS Euclidean and Hermitian self-dual codes over finite fields, Adv. in Pure Math., 7 (2017), 325-333.  doi: 10.4236/apm.2017.75019.  Google Scholar

[22]

H. Yan, A note on the constructions of MDS self-dual codes, Cryptogr. Commun., 11 (2019), 259-268.  doi: 10.1007/s12095-018-0288-3.  Google Scholar

[23]

A. Zhang and K. Feng, A unified approach to construct MDS self-dual codes via Reed-Solomon codes, IEEE Trans. Inform. Theory, 66 (2020), 3650-3656.  doi: 10.1109/TIT.2020.2963975.  Google Scholar

[24]

Z. Zhou, X. Li, C. Tang and C. Ding, Binary LCD codes and self-orthogonal codes from a generic construction, IEEE Trans. Inform. Theory, 65 (2019), 16-27.  doi: 10.1109/TIT.2018.2823704.  Google Scholar

show all references

References:
[1]

S. Balaji, M. Krishnan, M. Vajha, V. Ramkumar, B. Sasidharan and P. Kumar, Erasure coding for distributed storage: An overview, Sci. China Inf. Sci., 61 (2018), 100301.  doi: 10.1007/s11432-018-9482-6.  Google Scholar

[2]

J. H. Conway and N. J. A. Sloane, Sphere Packing, Lattices and Groups, 3$^rd$ edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-6568-7.  Google Scholar

[3]

R. Cramer, V. Daza, I. Gracia, J. J. Urroz, G. Leander, J. Marti-Farre and C. Padro, On codes, matroids and secure multiparty computation from linear secret sharing schemes, IEEE Trans. Inform. Theory, 54 (2008), 2644-2657.  doi: 10.1109/TIT.2008.921692.  Google Scholar

[4]

S. T. Dougherty, S. Mesnager and P. Sole, Secret-sharing schemes based on self-dual codes, IEEE Information Theory Workshop, (2008), 338–342. doi: 10.1109/ITW.2008.4578681.  Google Scholar

[5]

Z. Du, C. Li and S. Mesnager, Constructions of self-orthogonal codes from hulls of BCH codes and their parameters, IEEE Trans. Inform. Theory, 66 (2020), 6774-6785.  doi: 10.1109/TIT.2020.2991635.  Google Scholar

[6]

W. Fang and F.-W. Fu, New constructions of MDS Euclidean self-dual codes from GRS codes and extended GRS codes, IEEE Trans. Inform. Theory, 65 (2019), 5574-5579.  doi: 10.1109/TIT.2019.2916367.  Google Scholar

[7]

W. Fang, F.-W. Fu, L. Li and S. Zhu, Euclidean and Hermitian hulls of MDS codes and their applications to EAQECCs, IEEE Trans. Inform. Theory, 66 (2020), 3527-3537.  doi: 10.1109/TIT.2019.2950245.  Google Scholar

[8]

X. Fang, K. Lebed, H. Liu and J. Luo, New MDS self-dual codes over finite fields of odd characteristic, Des. Codes Cryptogr., 88 (2020), 1127-1138.  doi: 10.1007/s10623-020-00734-x.  Google Scholar

[9]

X. Fang, M. Liu and J. Luo, New MDS Euclidean self-orthogonal codes, IEEE Trans. Inform. Theory, 67 (2021), 130-137.  doi: 10.1109/TIT.2020.3020986.  Google Scholar

[10]

W. Fang, S.-T. Xia and F.-W. Fu, Construction of MDS Euclidean self-dual codes via two subsets, IEEE Trans. Inform. Theory, 67 (2021), 5005-5015.  doi: 10.1109/TIT.2021.3085768.  Google Scholar

[11]

W. Fang, J. Zhang, S.-T. Xia and F.-W. Fu, A note on self-dual generalized Reed-Solomon codes, preprint, arXiv: 2005.11732 [cs.IT]. Google Scholar

[12]

M. Grassl and T. A. Gulliver, On self-dual MDS codes, IEEE Int. Symp. Inform. Theory, (2008), 1954–1957. doi: 10.1109/ISIT.2008.4595330.  Google Scholar

[13]

M. Harada and H. Kharaghani, Orthogonal designs, self-dual codes, and the Leech lattice, J. Combin. Des., 13 (2005), 184-194.  doi: 10.1002/jcd.20046.  Google Scholar

[14]

M. Harada and H. Kharaghani, Orthogonal designs and MDS self-dual codes, Australas. J. Combin., 35 (2006), 57-67.   Google Scholar

[15]

A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays: Theory and Applications, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1478-6.  Google Scholar

[16] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[17]

L. Jin and C. 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

[18]

K. Lebed, H. Liu and J. Luo, Construction of MDS self-dual codes over finite fields, Finite Fields Appl., 59 (2019), 199-207.  doi: 10.1016/j.ffa.2019.05.007.  Google Scholar

[19]

R. Li, Z. Xu and X. Zhao, On the classification of binary optimal self-orthogonal codes, IEEE Trans. Inform. Theory, 54 (2008), 3778-3782.  doi: 10.1109/TIT.2008.926367.  Google Scholar

[20]

G. Luo, X. Cao and X. Chen, MDS codes with hulls of arbitrary dimensions and their quantum error correction, IEEE Trans. Inform. Theory, 65 (2019), 2944-2952.  doi: 10.1109/TIT.2018.2874953.  Google Scholar

[21]

H. Tong and X. Wang, New MDS Euclidean and Hermitian self-dual codes over finite fields, Adv. in Pure Math., 7 (2017), 325-333.  doi: 10.4236/apm.2017.75019.  Google Scholar

[22]

H. Yan, A note on the constructions of MDS self-dual codes, Cryptogr. Commun., 11 (2019), 259-268.  doi: 10.1007/s12095-018-0288-3.  Google Scholar

[23]

A. Zhang and K. Feng, A unified approach to construct MDS self-dual codes via Reed-Solomon codes, IEEE Trans. Inform. Theory, 66 (2020), 3650-3656.  doi: 10.1109/TIT.2020.2963975.  Google Scholar

[24]

Z. Zhou, X. Li, C. Tang and C. Ding, Binary LCD codes and self-orthogonal codes from a generic construction, IEEE Trans. Inform. Theory, 65 (2019), 16-27.  doi: 10.1109/TIT.2018.2823704.  Google Scholar

Table 1.  Some known results about MDS self-dual codes with length n
$ q $ $ n $ References
$ q $ even $ n\leq q $ [12]
$ q $ odd $ n=q+1 $ [12], [17]
$ q=r^{2} $ $ n\leq r $ [17]
$ q=r^{2} $, \; $ r\equiv3(mod\;4) $ $ n=2tr $, $ t\leq \frac{r-1}{2} $ [17]
$ q\equiv1(mod\;4) $ $ 4^{n}n^{2}\leq q $ [17]
$ q\equiv3(mod\;4) $ $ n\equiv 0 (mod\;4) $ and $ (n-1)\mid (q-1) $ [21]
$ q\equiv1(mod\;4) $ $ (n-1)\mid (q-1) $ [21]
$ q=p^{m}\equiv1(mod\;4) $ $ n=p^{l}+1 $, $ l\leq m $ [6]
$ q=r^{s} $, $ r $ odd, $ s $ even $ n=2tr^{l} $, $ 0 \leq l\leq s $ and $ 1 \leq t \leq\frac{r-1}{2} $ [6]
$ q=r^{s} $, $ r $ odd, $ s $ even $ n=(2t+1)r^{l}+1 $, $ 0 \leq l\leq s $ and $ 0 \leq t \leq\frac{r-1}{2} $ [6]
$ q $ odd $ (n-2) \mid (q-1) $, $ \eta(2-n)=1 $ [22], [6]
$ q $ odd $ (n-1) \mid (q-1) $, $ \eta(1-n)=1 $ [22]
$ q\equiv1(mod\;4) $ $ n \mid (q-1) $ [22]
$ q=p^{m} $, $ p $ odd $ n=p^{l}+1 $, $ l|m $ [22]
$ q=p^{m} $, $ p $ odd $ n=2p^{l} $, $ l<m $, $ \eta(-1)=1 $ [22]
$ q=r^{s} $, $ r $ odd, $ s\geq2 $ $ n=tr $, $ t $ even and $ 2t \mid (r-1) $ [22]
$ q=r^{s} $, $ r $ odd, $ s\geq2 $ $ n=tr $, $ t $ even, $ (t-1) \mid (r-1) $ and $ \eta(1-t)=1 $ [22]
$ q=r^{s} $, $ r $ odd, $ s\geq2 $ $ n=tr+1 $, $ t $ odd, $ t \mid (r-1) $ and $ \eta(t)=1 $ [22]
$ q=r^{s} $, $ r $ odd, $ s\geq2 $ $ n=tr+1 $, $ t $ odd, $ (t-1) \mid (r-1) $ and $ \eta(t-1)=\eta(-1)=1 $ [22]
$ q=r^{2} $, $ r $ odd $ n=tr $, $ t $ even, $ 1\leq t\leq r $ [22]
$ q=r^{2} $, $ r $ odd $ n=tr+1 $, $ t $ odd, $ 1\leq t\leq r $ [22]
$ q=r^{2} $, $ r $ odd $ n=tm $, $ \frac{q-1}{m} $ even and $ 1 \leq t \leq \frac{r+1}{\gcd(r+1,m)} $ [8]
$ q=r^{2} $, $ r $ odd $ n=tm+1 $, $ tm $ odd, $ m \mid (q-1) $ and $ 2 \leq t \leq \frac{r+1}{2\gcd(r+1,m)} $ [8]
$ q=r^{2} $, $ r $ odd $ n=tm+2 $, $ tm $ even, $ m \mid (q-1) $ (except $ t $, $ m $ are even and $ r\equiv1(mod\;4) $), and $ 1 \leq t \leq \frac{r+1}{\gcd(r+1,m)} $ [8]
$ q=r^{2} $, $ r $ odd $ n=tm $, $ \frac{q-1}{m} $ even, $ 1 \leq t \leq \frac{s(r-1)}{\gcd(s(r-1),m)} $, $ s $ even, $ s \mid m $ and $ \frac{r+1}{s} $ even [8]
$ q=r^{2} $, $ r $ odd $ n=tm+2 $, $ \frac{q-1}{m} $ even, $ \frac{r+1}{s} $ even, $ 1 \leq t \leq \frac{s(r-1)}{\gcd(s(r-1), m)} $, $ s $ even and $ s \mid m $ [8]
$ q=r^{2} $, $ r $ odd $ n=tm $, $ \frac{q-1}{m} $ even, $ 1 \leq t \leq \frac{r-1}{\gcd(r-1,m)} $ [18]
$ q=r^{2} $, $ r $ odd $ n=tm+1 $, $ tm $ odd, $ m \mid (q-1) $, $ 2 \leq t \leq \frac{r-1}{\gcd(r-1,m)} $ [18]
$ q=r^{2} $, $ r $ odd $ n=tm+2 $, $ tm $ even, $ m \mid (q-1) $, $ 2 \leq t \leq \frac{r-1}{\gcd(r-1,m)} $ [18]
$ q=r^{2} $, $ r\equiv1(mod\;4) $ $ n=s(r-1)+t(r+1) $, $ s $ even, $ 1 \leq s \leq \frac{r+1}{2} $ and $ 1 \leq t \leq \frac{r-1}{2} $ [9]
$ q=r^{2} $, $ r\equiv3(mod\;4) $ $ n=s(r-1)+t(r+1) $, $ s $ odd, $ 1 \leq s \leq \frac{r+1}{2} $ and $ 1 \leq t \leq \frac{r-1}{2} $ [9]
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$ q $ $ n $ References
$ q $ even $ n\leq q $ [12]
$ q $ odd $ n=q+1 $ [12], [17]
$ q=r^{2} $ $ n\leq r $ [17]
$ q=r^{2} $, \; $ r\equiv3(mod\;4) $ $ n=2tr $, $ t\leq \frac{r-1}{2} $ [17]
$ q\equiv1(mod\;4) $ $ 4^{n}n^{2}\leq q $ [17]
$ q\equiv3(mod\;4) $ $ n\equiv 0 (mod\;4) $ and $ (n-1)\mid (q-1) $ [21]
$ q\equiv1(mod\;4) $ $ (n-1)\mid (q-1) $ [21]
$ q=p^{m}\equiv1(mod\;4) $ $ n=p^{l}+1 $, $ l\leq m $ [6]
$ q=r^{s} $, $ r $ odd, $ s $ even $ n=2tr^{l} $, $ 0 \leq l\leq s $ and $ 1 \leq t \leq\frac{r-1}{2} $ [6]
$ q=r^{s} $, $ r $ odd, $ s $ even $ n=(2t+1)r^{l}+1 $, $ 0 \leq l\leq s $ and $ 0 \leq t \leq\frac{r-1}{2} $ [6]
$ q $ odd $ (n-2) \mid (q-1) $, $ \eta(2-n)=1 $ [22], [6]
$ q $ odd $ (n-1) \mid (q-1) $, $ \eta(1-n)=1 $ [22]
$ q\equiv1(mod\;4) $ $ n \mid (q-1) $ [22]
$ q=p^{m} $, $ p $ odd $ n=p^{l}+1 $, $ l|m $ [22]
$ q=p^{m} $, $ p $ odd $ n=2p^{l} $, $ l<m $, $ \eta(-1)=1 $ [22]
$ q=r^{s} $, $ r $ odd, $ s\geq2 $ $ n=tr $, $ t $ even and $ 2t \mid (r-1) $ [22]
$ q=r^{s} $, $ r $ odd, $ s\geq2 $ $ n=tr $, $ t $ even, $ (t-1) \mid (r-1) $ and $ \eta(1-t)=1 $ [22]
$ q=r^{s} $, $ r $ odd, $ s\geq2 $ $ n=tr+1 $, $ t $ odd, $ t \mid (r-1) $ and $ \eta(t)=1 $ [22]
$ q=r^{s} $, $ r $ odd, $ s\geq2 $ $ n=tr+1 $, $ t $ odd, $ (t-1) \mid (r-1) $ and $ \eta(t-1)=\eta(-1)=1 $ [22]
$ q=r^{2} $, $ r $ odd $ n=tr $, $ t $ even, $ 1\leq t\leq r $ [22]
$ q=r^{2} $, $ r $ odd $ n=tr+1 $, $ t $ odd, $ 1\leq t\leq r $ [22]
$ q=r^{2} $, $ r $ odd $ n=tm $, $ \frac{q-1}{m} $ even and $ 1 \leq t \leq \frac{r+1}{\gcd(r+1,m)} $ [8]
$ q=r^{2} $, $ r $ odd $ n=tm+1 $, $ tm $ odd, $ m \mid (q-1) $ and $ 2 \leq t \leq \frac{r+1}{2\gcd(r+1,m)} $ [8]
$ q=r^{2} $, $ r $ odd $ n=tm+2 $, $ tm $ even, $ m \mid (q-1) $ (except $ t $, $ m $ are even and $ r\equiv1(mod\;4) $), and $ 1 \leq t \leq \frac{r+1}{\gcd(r+1,m)} $ [8]
$ q=r^{2} $, $ r $ odd $ n=tm $, $ \frac{q-1}{m} $ even, $ 1 \leq t \leq \frac{s(r-1)}{\gcd(s(r-1),m)} $, $ s $ even, $ s \mid m $ and $ \frac{r+1}{s} $ even [8]
$ q=r^{2} $, $ r $ odd $ n=tm+2 $, $ \frac{q-1}{m} $ even, $ \frac{r+1}{s} $ even, $ 1 \leq t \leq \frac{s(r-1)}{\gcd(s(r-1), m)} $, $ s $ even and $ s \mid m $ [8]
$ q=r^{2} $, $ r $ odd $ n=tm $, $ \frac{q-1}{m} $ even, $ 1 \leq t \leq \frac{r-1}{\gcd(r-1,m)} $ [18]
$ q=r^{2} $, $ r $ odd $ n=tm+1 $, $ tm $ odd, $ m \mid (q-1) $, $ 2 \leq t \leq \frac{r-1}{\gcd(r-1,m)} $ [18]
$ q=r^{2} $, $ r $ odd $ n=tm+2 $, $ tm $ even, $ m \mid (q-1) $, $ 2 \leq t \leq \frac{r-1}{\gcd(r-1,m)} $ [18]
$ q=r^{2} $, $ r\equiv1(mod\;4) $ $ n=s(r-1)+t(r+1) $, $ s $ even, $ 1 \leq s \leq \frac{r+1}{2} $ and $ 1 \leq t \leq \frac{r-1}{2} $ [9]
$ q=r^{2} $, $ r\equiv3(mod\;4) $ $ n=s(r-1)+t(r+1) $, $ s $ odd, $ 1 \leq s \leq \frac{r+1}{2} $ and $ 1 \leq t \leq \frac{r-1}{2} $ [9]
Table 2.  Proportion of number of possible lengths to $\frac{q}{2}$ ($N$ is the number of possible lengths)
$r$ $q$ $N/(\frac{q}{2}$) of Table 1 (except [9]) $N/(\frac{q}{2})$ of [9] $N/\frac{q}{2}$ of us number of new lengths
149 22201 $11.89\%$ $25\%$ $38.61\%$ 775
151 22801 $13.16\%$ $25\%$ $34.95\%$ 676
157 24649 $10.18\%$ $25\%$ $34.95\%$ 758
163 26569 $10.67\%$ $25\%$ $34.28\%$ 828
167 27889 $13.90\%$ $25\%$ $34.27\%$ 704
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$r$ $q$ $N/(\frac{q}{2}$) of Table 1 (except [9]) $N/(\frac{q}{2})$ of [9] $N/\frac{q}{2}$ of us number of new lengths
149 22201 $11.89\%$ $25\%$ $38.61\%$ 775
151 22801 $13.16\%$ $25\%$ $34.95\%$ 676
157 24649 $10.18\%$ $25\%$ $34.95\%$ 758
163 26569 $10.67\%$ $25\%$ $34.28\%$ 828
167 27889 $13.90\%$ $25\%$ $34.27\%$ 704
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