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doi: 10.3934/amc.2020120

Three classes of partitioned difference families and their optimal constant composition codes

1. 

College of Liberal Arts and Science, National University of Defense Technology, Changsha 410073, China

2. 

Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing 211167, China

3. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received  February 2020 Revised  August 2020 Published  December 2020

Cyclotomy, firstly introduced by Gauss, is an important topic in Mathematics since it has a number of applications in number theory, combinatorics, coding theory and cryptography. Depending on $ v $ prime or composite, cyclotomy on a residue class ring $ {\mathbb{Z}}_{v} $ can be divided into classical cyclotomy or generalized cyclotomy. Inspired by a foregoing work of Zeng et al. [40], we introduce a generalized cyclotomy of order $ e $ on the ring $ {\rm GF}(q_1)\times {\rm GF}(q_2)\times \cdots \times {\rm GF}(q_k) $, where $ q_i $ and $ q_j $ ($ i\neq j $) may not be co-prime, which includes classical cyclotomy as a special case. Here, $ q_1 $, $ q_2 $, $ \cdots $, $ q_k $ are powers of primes with an integer $ e|(q_i-1) $ for any $ 1\leq i\leq k $. Then we obtain some basic properties of the corresponding generalized cyclotomic numbers. Furthermore, we propose three classes of partitioned difference families by means of the generalized cyclotomy above and $ d $-form functions with difference balanced property. Afterwards, three families of optimal constant composition codes from these partitioned difference families are obtained, and their parameters are also summarized.

Citation: Shanding Xu, Longjiang Qu, Xiwang Cao. Three classes of partitioned difference families and their optimal constant composition codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2020120
References:
[1]

K. ArasuJ. Dillon and K. Player, Character sum factorizations yield sequences with ideal two-level autocorrelation, IEEE Trans. Inf. Theory, 61 (2015), 3276-3304.  doi: 10.1109/TIT.2015.2418204.  Google Scholar

[2]

M. Buratti, Hadamard partitioned difference families and their descendants, Cryptogr. Commun., 11 (2019), 557-562.  doi: 10.1007/s12095-018-0308-3.  Google Scholar

[3]

M. Buratti, On disjoint $(v, k, k-1)$ difference families, Des. Codes Cryptogr., 87 (2019), 745-755.  doi: 10.1007/s10623-018-0511-4.  Google Scholar

[4]

M. Buratti and D. Jungnickel, Partitioned difference families versus zero-difference balanced functions, Des. Codes Cryptogr., 87 (2019), 2461-2467.  doi: 10.1007/s10623-019-00632-x.  Google Scholar

[5]

M. Buratti, J. Yan and C. Wang, From a 1-rotational RBIBD to a partitioned difference family, Electron. J. Comb., 17 (2010), pp. R139.  Google Scholar

[6]

H. CaiZ. ZhouX. Tang and Y. Miao, Zero-difference balanced functions with new parameters and their applications, IEEE Trans. Inf. Theory, 63 (2017), 4379-4387.  doi: 10.1109/TIT.2017.2675441.  Google Scholar

[7]

Y. Chang and C. Ding, Constructions of external difference families and disjoint difference families, Des. Codes Cryptogr., 40 (2006), 167-185.  doi: 10.1007/s10623-006-0005-7.  Google Scholar

[8]

W. Chu and C. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inf. Theory, 51 (2005), 1139-1141.  doi: 10.1109/TIT.2004.842708.  Google Scholar

[9]

J. Chung and K. Yang, $k$-fold cyclotomy and its application to frequency-hopping sequences, IEEE Trans. Inf. Theory, 57 (2011), 2306-2317.  doi: 10.1109/TIT.2011.2112235.  Google Scholar

[10]

C. Ding, Cyclic codes from cyclotomic sequences of order four, Finite Fields Appl., 23 (2013), 8-34.  doi: 10.1016/j.ffa.2013.03.006.  Google Scholar

[11]

C. Ding, Optimal constant composition codes from zero-difference balanced functions, IEEE Trans. Inf. Theory, 54 (2008), 5766-5770.  doi: 10.1109/TIT.2008.2006420.  Google Scholar

[12]

C. Ding, Optimal and perfect difference systems of sets, J. Comb. Theory, Series A, 116 (2009), 109-119.  doi: 10.1016/j.jcta.2008.05.007.  Google Scholar

[13]

C. Ding and Y. Tan, Zero-difference balanced functions with applications, J. Stat. Theory and Practice, 6 (2012), 3-19.  doi: 10.1080/15598608.2012.647479.  Google Scholar

[14]

C. Ding and J. Yin, Combinatorial constructions of optimal constant-composition codes, IEEE Trans. Inf. Theory, 51 (2005), 3671-3674.  doi: 10.1109/TIT.2005.855612.  Google Scholar

[15]

C. Ding and T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl., 4 (1998), 140-166.  doi: 10.1006/ffta.1998.0207.  Google Scholar

[16]

C. Ding and T. Helleseth, Generalized cyclotomic codes of length $p_{1}^{e_{1}}\cdots p_{t}^{e_{t}}$, IEEE Trans. Inf. Theory, 45 (1999), 467-474.  doi: 10.1109/18.748996.  Google Scholar

[17]

C. Fan and G. Ge, A unified approach to Whiteman's and Ding-Helleseth's generalized cyclotomy over residue class rings, IEEE Trans. Inf. Theory, 60 (2014), 1326-1336.  doi: 10.1109/TIT.2013.2290694.  Google Scholar

[18]

R. Fuji-HaraY. Miao and M. Mishima, Optimal frequency hopping sequences: A combinatorial approach, IEEE Trans. Inf. Theory, 50 (2004), 2408-2420.  doi: 10.1109/TIT.2004.834783.  Google Scholar

[19]

C. F. Gauss, Disquisitiones Arithmeticae, New York, USA: Springer-Verlag, 1986.  Google Scholar

[20]

G. GeY. Miao and Z. Yao, Optimal frequency hopping sequences: Auto- and cross-correlation properties, IEEE Trans. Inf. Theory, 55 (2009), 867-879.  doi: 10.1109/TIT.2008.2009856.  Google Scholar

[21]

B. GordonW. H. Mills and L. R. Welch, Some new difference sets, Canad. J. Math., 14 (1962), 614-625.  doi: 10.4153/CJM-1962-052-2.  Google Scholar

[22]

Y. Han and K. Yang, On the Sidel'nikov sequences as frequency-hopping sequences, IEEE Trans. Inf. Theory, 55 (2009), 4279-4285.  doi: 10.1109/TIT.2009.2025569.  Google Scholar

[23]

T. Helleseth and G. Gong, New nonbinary sequences with ideal two-level autocorrelation, IEEE Trans. Inf. Theory, 48 (2002), 2868-2872.  doi: 10.1109/TIT.2002.804052.  Google Scholar

[24]

H. HuS. ShaoG. Gong and T. Helleseth, The proof of Lin's conjecture via the decimation-Hadamard transform, IEEE Trans. Inf. Theory, 60 (2013), 5054-5064.  doi: 10.1109/TIT.2014.2327625.  Google Scholar

[25]

L. Hu and Q. Yue, Gauss periods and codebooks from generalized cyclotomic sets of order four, Des. Codes Cryptogr., 69 (2013), 233-246.  doi: 10.1007/s10623-012-9648-8.  Google Scholar

[26]

A. Klapper, $d$-form sequence: Families of sequences with low correlaltion values and large linear spans, IEEE Trans. Inf. Theory, 51 (1995), 1469-1477.  doi: 10.1109/18.370143.  Google Scholar

[27]

S. LiH. Wei and G. Ge, Generic constructions for partitioned difference families with applications: A unified combinatorial approach, Des. Codes Cryptogr., 82 (2017), 583-599.  doi: 10.1007/s10623-016-0182-y.  Google Scholar

[28]

H. A. Lin, From cyclic Hadamard difference sets to perfectly balanced sequences, Ph.D. thesis, University of Southern California, 1998. Google Scholar

[29]

J. LiuY. JiangQ. Zheng and D. Lin, A new construction of zero-difference balanced functions and two applications, Des. Codes Cryptogr., 87 (2019), 2251-2265.  doi: 10.1007/s10623-019-00616-x.  Google Scholar

[30]

Y. LuoF. FuA. Vinck and W. Chen, On constant-composition codes over ${{\mathbb{Z}}_{q}}$, IEEE Trans. Inf. Theory, 49 (2003), 3010-3016.  doi: 10.1109/TIT.2003.819339.  Google Scholar

[31]

J.-S. No, New cyclic difference sets with Singer parameters constructed from $d$-homogeneous functions, Des. Codes Cryptogr., 33 (2004), 199-213.  doi: 10.1023/B:DESI.0000036246.52472.81.  Google Scholar

[32]

T. Storer, Cyclotomy and Difference Sets, Chicago: Markham Pub. Co., 1967.  Google Scholar

[33]

Q. Wang and Y. Zhou, Sets of zero-difference balanced functions and their applications, Adv. Math. Commun., 8 (2014), 83-101.  doi: 10.3934/amc.2014.8.83.  Google Scholar

[34]

X. Wang and J. Wang, Partitioned difference families and almost difference sets, J. Stat. Plan. Inference, 141 (2011), 1899-1909.  doi: 10.1016/j.jspi.2010.12.002.  Google Scholar

[35]

A. L. Whiteman, A family of difference sets, Illinois J. Math., 6 (1962), 107-121.  doi: 10.1215/ijm/1255631810.  Google Scholar

[36]

R. M. Wilson, Cyclotomic and difference families in elementary abelian groups, J. Number Theory, 4 (1972), 17-47.  doi: 10.1016/0022-314X(72)90009-1.  Google Scholar

[37]

Y. Yang, Z. Zhou and X. Tang, Two classes of zero-difference balanced functions and their optimal constant composition codes, in Proceedings of 2016 IEEE International Symposium on Information Theory, (2016), 1327–1330. doi: 10.1109/TIT.2008.2006420.  Google Scholar

[38]

Z. YiZ. Lin and L. Ke, A generic method to construct zero-difference balanced functions, Cryptogr. Commun., 10 (2018), 591-609.  doi: 10.1007/s12095-017-0247-4.  Google Scholar

[39]

J. YinX. Shan and Z. Tian, Constructions of partitioned difference families, Eur. J. Comb., 29 (2008), 1507-1519.  doi: 10.1016/j.ejc.2007.06.006.  Google Scholar

[40]

X. ZengH. CaiX. Tang and Y. Yang, Optimal frequency hopping sequences of odd length, IEEE Trans. Inf. Theory, 59 (2013), 3237-3248.  doi: 10.1109/TIT.2013.2237754.  Google Scholar

[41]

Z. Zha and L. Hu, Cyclotomic constructions of zero-difference balanced functions with applications, IEEE Trans. Inf. Theory, 61 (2015), 1491–1495. doi: 10.1109/TIT.2014.2388231.  Google Scholar

[42]

Z. ZhouX. TangD. Wu and Y. Yang, Some new classes of zero-difference balanced functions, IEEE Trans. Inf. Theory, 58 (2012), 139-145.  doi: 10.1109/TIT.2011.2171418.  Google Scholar

show all references

References:
[1]

K. ArasuJ. Dillon and K. Player, Character sum factorizations yield sequences with ideal two-level autocorrelation, IEEE Trans. Inf. Theory, 61 (2015), 3276-3304.  doi: 10.1109/TIT.2015.2418204.  Google Scholar

[2]

M. Buratti, Hadamard partitioned difference families and their descendants, Cryptogr. Commun., 11 (2019), 557-562.  doi: 10.1007/s12095-018-0308-3.  Google Scholar

[3]

M. Buratti, On disjoint $(v, k, k-1)$ difference families, Des. Codes Cryptogr., 87 (2019), 745-755.  doi: 10.1007/s10623-018-0511-4.  Google Scholar

[4]

M. Buratti and D. Jungnickel, Partitioned difference families versus zero-difference balanced functions, Des. Codes Cryptogr., 87 (2019), 2461-2467.  doi: 10.1007/s10623-019-00632-x.  Google Scholar

[5]

M. Buratti, J. Yan and C. Wang, From a 1-rotational RBIBD to a partitioned difference family, Electron. J. Comb., 17 (2010), pp. R139.  Google Scholar

[6]

H. CaiZ. ZhouX. Tang and Y. Miao, Zero-difference balanced functions with new parameters and their applications, IEEE Trans. Inf. Theory, 63 (2017), 4379-4387.  doi: 10.1109/TIT.2017.2675441.  Google Scholar

[7]

Y. Chang and C. Ding, Constructions of external difference families and disjoint difference families, Des. Codes Cryptogr., 40 (2006), 167-185.  doi: 10.1007/s10623-006-0005-7.  Google Scholar

[8]

W. Chu and C. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inf. Theory, 51 (2005), 1139-1141.  doi: 10.1109/TIT.2004.842708.  Google Scholar

[9]

J. Chung and K. Yang, $k$-fold cyclotomy and its application to frequency-hopping sequences, IEEE Trans. Inf. Theory, 57 (2011), 2306-2317.  doi: 10.1109/TIT.2011.2112235.  Google Scholar

[10]

C. Ding, Cyclic codes from cyclotomic sequences of order four, Finite Fields Appl., 23 (2013), 8-34.  doi: 10.1016/j.ffa.2013.03.006.  Google Scholar

[11]

C. Ding, Optimal constant composition codes from zero-difference balanced functions, IEEE Trans. Inf. Theory, 54 (2008), 5766-5770.  doi: 10.1109/TIT.2008.2006420.  Google Scholar

[12]

C. Ding, Optimal and perfect difference systems of sets, J. Comb. Theory, Series A, 116 (2009), 109-119.  doi: 10.1016/j.jcta.2008.05.007.  Google Scholar

[13]

C. Ding and Y. Tan, Zero-difference balanced functions with applications, J. Stat. Theory and Practice, 6 (2012), 3-19.  doi: 10.1080/15598608.2012.647479.  Google Scholar

[14]

C. Ding and J. Yin, Combinatorial constructions of optimal constant-composition codes, IEEE Trans. Inf. Theory, 51 (2005), 3671-3674.  doi: 10.1109/TIT.2005.855612.  Google Scholar

[15]

C. Ding and T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl., 4 (1998), 140-166.  doi: 10.1006/ffta.1998.0207.  Google Scholar

[16]

C. Ding and T. Helleseth, Generalized cyclotomic codes of length $p_{1}^{e_{1}}\cdots p_{t}^{e_{t}}$, IEEE Trans. Inf. Theory, 45 (1999), 467-474.  doi: 10.1109/18.748996.  Google Scholar

[17]

C. Fan and G. Ge, A unified approach to Whiteman's and Ding-Helleseth's generalized cyclotomy over residue class rings, IEEE Trans. Inf. Theory, 60 (2014), 1326-1336.  doi: 10.1109/TIT.2013.2290694.  Google Scholar

[18]

R. Fuji-HaraY. Miao and M. Mishima, Optimal frequency hopping sequences: A combinatorial approach, IEEE Trans. Inf. Theory, 50 (2004), 2408-2420.  doi: 10.1109/TIT.2004.834783.  Google Scholar

[19]

C. F. Gauss, Disquisitiones Arithmeticae, New York, USA: Springer-Verlag, 1986.  Google Scholar

[20]

G. GeY. Miao and Z. Yao, Optimal frequency hopping sequences: Auto- and cross-correlation properties, IEEE Trans. Inf. Theory, 55 (2009), 867-879.  doi: 10.1109/TIT.2008.2009856.  Google Scholar

[21]

B. GordonW. H. Mills and L. R. Welch, Some new difference sets, Canad. J. Math., 14 (1962), 614-625.  doi: 10.4153/CJM-1962-052-2.  Google Scholar

[22]

Y. Han and K. Yang, On the Sidel'nikov sequences as frequency-hopping sequences, IEEE Trans. Inf. Theory, 55 (2009), 4279-4285.  doi: 10.1109/TIT.2009.2025569.  Google Scholar

[23]

T. Helleseth and G. Gong, New nonbinary sequences with ideal two-level autocorrelation, IEEE Trans. Inf. Theory, 48 (2002), 2868-2872.  doi: 10.1109/TIT.2002.804052.  Google Scholar

[24]

H. HuS. ShaoG. Gong and T. Helleseth, The proof of Lin's conjecture via the decimation-Hadamard transform, IEEE Trans. Inf. Theory, 60 (2013), 5054-5064.  doi: 10.1109/TIT.2014.2327625.  Google Scholar

[25]

L. Hu and Q. Yue, Gauss periods and codebooks from generalized cyclotomic sets of order four, Des. Codes Cryptogr., 69 (2013), 233-246.  doi: 10.1007/s10623-012-9648-8.  Google Scholar

[26]

A. Klapper, $d$-form sequence: Families of sequences with low correlaltion values and large linear spans, IEEE Trans. Inf. Theory, 51 (1995), 1469-1477.  doi: 10.1109/18.370143.  Google Scholar

[27]

S. LiH. Wei and G. Ge, Generic constructions for partitioned difference families with applications: A unified combinatorial approach, Des. Codes Cryptogr., 82 (2017), 583-599.  doi: 10.1007/s10623-016-0182-y.  Google Scholar

[28]

H. A. Lin, From cyclic Hadamard difference sets to perfectly balanced sequences, Ph.D. thesis, University of Southern California, 1998. Google Scholar

[29]

J. LiuY. JiangQ. Zheng and D. Lin, A new construction of zero-difference balanced functions and two applications, Des. Codes Cryptogr., 87 (2019), 2251-2265.  doi: 10.1007/s10623-019-00616-x.  Google Scholar

[30]

Y. LuoF. FuA. Vinck and W. Chen, On constant-composition codes over ${{\mathbb{Z}}_{q}}$, IEEE Trans. Inf. Theory, 49 (2003), 3010-3016.  doi: 10.1109/TIT.2003.819339.  Google Scholar

[31]

J.-S. No, New cyclic difference sets with Singer parameters constructed from $d$-homogeneous functions, Des. Codes Cryptogr., 33 (2004), 199-213.  doi: 10.1023/B:DESI.0000036246.52472.81.  Google Scholar

[32]

T. Storer, Cyclotomy and Difference Sets, Chicago: Markham Pub. Co., 1967.  Google Scholar

[33]

Q. Wang and Y. Zhou, Sets of zero-difference balanced functions and their applications, Adv. Math. Commun., 8 (2014), 83-101.  doi: 10.3934/amc.2014.8.83.  Google Scholar

[34]

X. Wang and J. Wang, Partitioned difference families and almost difference sets, J. Stat. Plan. Inference, 141 (2011), 1899-1909.  doi: 10.1016/j.jspi.2010.12.002.  Google Scholar

[35]

A. L. Whiteman, A family of difference sets, Illinois J. Math., 6 (1962), 107-121.  doi: 10.1215/ijm/1255631810.  Google Scholar

[36]

R. M. Wilson, Cyclotomic and difference families in elementary abelian groups, J. Number Theory, 4 (1972), 17-47.  doi: 10.1016/0022-314X(72)90009-1.  Google Scholar

[37]

Y. Yang, Z. Zhou and X. Tang, Two classes of zero-difference balanced functions and their optimal constant composition codes, in Proceedings of 2016 IEEE International Symposium on Information Theory, (2016), 1327–1330. doi: 10.1109/TIT.2008.2006420.  Google Scholar

[38]

Z. YiZ. Lin and L. Ke, A generic method to construct zero-difference balanced functions, Cryptogr. Commun., 10 (2018), 591-609.  doi: 10.1007/s12095-017-0247-4.  Google Scholar

[39]

J. YinX. Shan and Z. Tian, Constructions of partitioned difference families, Eur. J. Comb., 29 (2008), 1507-1519.  doi: 10.1016/j.ejc.2007.06.006.  Google Scholar

[40]

X. ZengH. CaiX. Tang and Y. Yang, Optimal frequency hopping sequences of odd length, IEEE Trans. Inf. Theory, 59 (2013), 3237-3248.  doi: 10.1109/TIT.2013.2237754.  Google Scholar

[41]

Z. Zha and L. Hu, Cyclotomic constructions of zero-difference balanced functions with applications, IEEE Trans. Inf. Theory, 61 (2015), 1491–1495. doi: 10.1109/TIT.2014.2388231.  Google Scholar

[42]

Z. ZhouX. TangD. Wu and Y. Yang, Some new classes of zero-difference balanced functions, IEEE Trans. Inf. Theory, 58 (2012), 139-145.  doi: 10.1109/TIT.2011.2171418.  Google Scholar

Table 1.  $ (A, K, \lambda) $ PDF constructed in this paper
$ A $ $ K $ $ \lambda $ Constraints Ref.
$ R\times {\mathbb{Z}}_{e} $ $ [{(e-1)}^{\frac{ev-1}{e-1}}1^{1}] $ $ e-2 $ $ v=q_{1} q_{2} \cdots q_{k} $, $ e(e-1)|(q_i-1) $ for $ 1\leq i\leq k $ Theorem 3.4
$ R $ $ [e^{\frac{v-1}{2e}}1^{\frac{v+1}{2}}] $ $ \frac{e-1}{2} $ $ v=q_{1} q_{2} \cdots q_{k} $, $ e\geq 3 $ is odd such that $ e|(q_i-1) $ for $ 1\leq i\leq k $ Theorem 3.6
$ {\mathbb{Z}}_{\frac{q^m-1}{e}} \times {\mathbb{Z}}_{k} $ $ [k\frac{q^{m-1}-1}{e}^{1}1^{k \frac{q^m-q^{m-1}}{e}}] $ $ k \frac{q^{m-2}-1}{e} $ $ e |(q-1), \operatorname{gcd}(e, m)=1 $, $ 1 \leq k \leq e $, $ m>2 $ Theorem 3.9
$ A $ $ K $ $ \lambda $ Constraints Ref.
$ R\times {\mathbb{Z}}_{e} $ $ [{(e-1)}^{\frac{ev-1}{e-1}}1^{1}] $ $ e-2 $ $ v=q_{1} q_{2} \cdots q_{k} $, $ e(e-1)|(q_i-1) $ for $ 1\leq i\leq k $ Theorem 3.4
$ R $ $ [e^{\frac{v-1}{2e}}1^{\frac{v+1}{2}}] $ $ \frac{e-1}{2} $ $ v=q_{1} q_{2} \cdots q_{k} $, $ e\geq 3 $ is odd such that $ e|(q_i-1) $ for $ 1\leq i\leq k $ Theorem 3.6
$ {\mathbb{Z}}_{\frac{q^m-1}{e}} \times {\mathbb{Z}}_{k} $ $ [k\frac{q^{m-1}-1}{e}^{1}1^{k \frac{q^m-q^{m-1}}{e}}] $ $ k \frac{q^{m-2}-1}{e} $ $ e |(q-1), \operatorname{gcd}(e, m)=1 $, $ 1 \leq k \leq e $, $ m>2 $ Theorem 3.9
Table 2.  Some optimal CCCs with parameters $ (n, M, d, [\omega_0, \omega_1, \cdots, \omega_{m-1}])_m $ from our PDFs
Parameters Constraints
$ (e v, e v, e v-e+2, [{(e-1)}^{\frac{ev-1}{e-1}}1^{1}])_\frac{ev+e-2}{e-1} $ $ v=q_{1} q_{2} \cdots q_{k} $, $ e(e-1)|(q_i-1) $ for $ 1\leq i\leq k $
$ \left(v, v, v-\frac{e-1}{2}, [e^{\frac{v-1}{2e}}1^{\frac{v+1}{2}}]\right)_{\frac{v-1}{2 e}+\frac{v+1}{2}} $ $ v=q_{1} q_{2} \cdots q_{k} $, $ e\geq 3 $ is odd such that $ e|(q_i-1) $ for $ 1\leq i\leq k $
$ \left(k \frac{q^{m}-1}{e}, k \frac{q^{m}-1}{e}, k \frac{q^{m-2}-1}{e}, [1^{k \frac{q^m-q^{m-1}}{e}} k\frac{q^{m-1}-1}{e}^{1}]\right)_{k \frac{q^m-q^{m-1}}{e}+1} $ $ e|(q-1), \gcd(e, m)=1, $ $ 1\leq k\leq e, m>2 $
Parameters Constraints
$ (e v, e v, e v-e+2, [{(e-1)}^{\frac{ev-1}{e-1}}1^{1}])_\frac{ev+e-2}{e-1} $ $ v=q_{1} q_{2} \cdots q_{k} $, $ e(e-1)|(q_i-1) $ for $ 1\leq i\leq k $
$ \left(v, v, v-\frac{e-1}{2}, [e^{\frac{v-1}{2e}}1^{\frac{v+1}{2}}]\right)_{\frac{v-1}{2 e}+\frac{v+1}{2}} $ $ v=q_{1} q_{2} \cdots q_{k} $, $ e\geq 3 $ is odd such that $ e|(q_i-1) $ for $ 1\leq i\leq k $
$ \left(k \frac{q^{m}-1}{e}, k \frac{q^{m}-1}{e}, k \frac{q^{m-2}-1}{e}, [1^{k \frac{q^m-q^{m-1}}{e}} k\frac{q^{m-1}-1}{e}^{1}]\right)_{k \frac{q^m-q^{m-1}}{e}+1} $ $ e|(q-1), \gcd(e, m)=1, $ $ 1\leq k\leq e, m>2 $
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