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On the equivalence of several classes of quaternary sequences with optimal autocorrelation and length $ 2p$
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 |
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. [
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K. Arasu, J. 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. |
[2] |
M. Buratti,
Hadamard partitioned difference families and their descendants, Cryptogr. Commun., 11 (2019), 557-562.
doi: 10.1007/s12095-018-0308-3. |
[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. |
[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. |
[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. |
[6] |
H. Cai, Z. Zhou, X. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
R. Fuji-Hara, Y. 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. |
[19] |
C. F. Gauss, Disquisitiones Arithmeticae, New York, USA: Springer-Verlag, 1986. |
[20] |
G. Ge, Y. 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. |
[21] |
B. Gordon, W. H. Mills and L. R. Welch,
Some new difference sets, Canad. J. Math., 14 (1962), 614-625.
doi: 10.4153/CJM-1962-052-2. |
[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. |
[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. |
[24] |
H. Hu, S. Shao, G. 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. |
[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. |
[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. |
[27] |
S. Li, H. 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. |
[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. Liu, Y. Jiang, Q. 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. |
[30] |
Y. Luo, F. Fu, A. 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. |
[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. |
[32] |
T. Storer, Cyclotomy and Difference Sets, Chicago: Markham Pub. Co., 1967. |
[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. |
[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. |
[35] |
A. L. Whiteman,
A family of difference sets, Illinois J. Math., 6 (1962), 107-121.
doi: 10.1215/ijm/1255631810. |
[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. |
[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. |
[38] |
Z. Yi, Z. 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. |
[39] |
J. Yin, X. Shan and Z. Tian,
Constructions of partitioned difference families, Eur. J. Comb., 29 (2008), 1507-1519.
doi: 10.1016/j.ejc.2007.06.006. |
[40] |
X. Zeng, H. Cai, X. 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. |
[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. |
[42] |
Z. Zhou, X. Tang, D. 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. |
show all references
References:
[1] |
K. Arasu, J. 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. |
[2] |
M. Buratti,
Hadamard partitioned difference families and their descendants, Cryptogr. Commun., 11 (2019), 557-562.
doi: 10.1007/s12095-018-0308-3. |
[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. |
[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. |
[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. |
[6] |
H. Cai, Z. Zhou, X. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
R. Fuji-Hara, Y. 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. |
[19] |
C. F. Gauss, Disquisitiones Arithmeticae, New York, USA: Springer-Verlag, 1986. |
[20] |
G. Ge, Y. 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. |
[21] |
B. Gordon, W. H. Mills and L. R. Welch,
Some new difference sets, Canad. J. Math., 14 (1962), 614-625.
doi: 10.4153/CJM-1962-052-2. |
[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. |
[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. |
[24] |
H. Hu, S. Shao, G. 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. |
[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. |
[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. |
[27] |
S. Li, H. 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. |
[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. Liu, Y. Jiang, Q. 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. |
[30] |
Y. Luo, F. Fu, A. 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. |
[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. |
[32] |
T. Storer, Cyclotomy and Difference Sets, Chicago: Markham Pub. Co., 1967. |
[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. |
[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. |
[35] |
A. L. Whiteman,
A family of difference sets, Illinois J. Math., 6 (1962), 107-121.
doi: 10.1215/ijm/1255631810. |
[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. |
[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. |
[38] |
Z. Yi, Z. 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. |
[39] |
J. Yin, X. Shan and Z. Tian,
Constructions of partitioned difference families, Eur. J. Comb., 29 (2008), 1507-1519.
doi: 10.1016/j.ejc.2007.06.006. |
[40] |
X. Zeng, H. Cai, X. 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. |
[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. |
[42] |
Z. Zhou, X. Tang, D. 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. |
Constraints | Ref. | |||
Theorem 3.4 | ||||
Theorem 3.6 | ||||
Theorem 3.9 |
Constraints | Ref. | |||
Theorem 3.4 | ||||
Theorem 3.6 | ||||
Theorem 3.9 |
Parameters | Constraints |
|
|
|
|
Parameters | Constraints |
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