doi: 10.3934/amc.2021010

Partial direct product difference sets and almost quaternary sequences

1. 

Hacettepe University, Graduate School of Science and Engineering, Beytepe, Ankara, Turkey

2. 

Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey

* Corresponding author

This paper is a part of Büşra Özden's PhD thesis

Received  May 2020 Revised  January 2021 Published  May 2021

In this paper, we study the $ m $-ary sequences with (non-consecutive) two zero-symbols and at most two distinct autocorrelation coefficients, which are known as almost $ m $-ary nearly perfect sequences. We show that these sequences are equivalent to $ \ell $-partial direct product difference sets (PDPDS), then we extend known results on the sequences with two consecutive zero-symbols to non-consecutive case. Next, we study the notion of multipliers and orbit combination for $ \ell $-PDPDS. Finally, we present two construction methods for a family of almost quaternary sequences with at most two out-of-phase autocorrelation coefficients.

Citation: Büşra Özden, Oǧuz Yayla. Partial direct product difference sets and almost quaternary sequences. Advances in Mathematics of Communications, doi: 10.3934/amc.2021010
References:
[1]

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

[2]

K. T. ArasuC. DingT. HellesethP. V. Kumar and H. M. Martinsen, Almost difference sets and their sequences with optimal autocorrelation, IEEE Transactions on Information Theory, 47 (2001), 2934-2943.  doi: 10.1109/18.959271.  Google Scholar

[3] T. BethD. Jungnickel and H. Lenz, Design Theory: Volume 1, Cambridge University Press, 1999.   Google Scholar
[4]

Y. Cai and C. Ding, Binary sequences with optimal autocorrelation, Theoretical Computer Science, 410 (2009), 2316-2322.  doi: 10.1016/j.tcs.2009.02.021.  Google Scholar

[5]

A. Çeșmelioǧlu and O. Olmez, Graphs of vectorial plateaued functions as difference sets, Finite Fields and Their Applications, 71 (2021), 101795. doi: 10.1016/j.ffa.2020.101795.  Google Scholar

[6]

Y. M. Chee, Y. Tan and Y. Zhou, Almost p-ary perfect sequences, in International Conference on Sequences and Their Applications doi: 10.1007/978-3-642-15874-2_34.  Google Scholar

[7]

I. Chih-Lin and R. D. Gitlin, Multi-code CDMA wireless personal communications networks, Proceedings IEEE International Conference on Communications ICC'95, 2 (1995), 1060-1064.  doi: 10.1109/ICC.1995.524263.  Google Scholar

[8] C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, CRC press, 2006.  doi: 10.1201/9781420049954.  Google Scholar
[9]

L. E. Dickson, Cyclotomy, higher congruences, and Waring's problem, American Journal of Mathematics, 57 (1935), 391-424.  doi: 10.2307/2371217.  Google Scholar

[10]

J. F. Dillon and H. Dobbertin, New cyclic difference sets with singer parameters, Finite Fields and Their Applications, 10 (2004), 342-389.  doi: 10.1016/j.ffa.2003.09.003.  Google Scholar

[11]

C. DingT. Helleseth and K. Y. Lam, Several classes of binary sequences with three-level autocorrelation, IEEE Transactions on Information Theory, 45 (1999), 2606-2612.  doi: 10.1109/18.796414.  Google Scholar

[12]

C. DingT. Helleseth and H. Martinsen, New families of binary sequences with optimal three-level autocorrelation, IEEE Transactions on Information Theory, 47 (2001), 428-433.  doi: 10.1109/18.904555.  Google Scholar

[13]

V. E. Gantmakher and M. V. Zaleshin, Almost six-phase sequences with perfect periodic autocorrelation function, in International Conference on Sequences and Their Applications doi: 10.1007/978-3-319-12325-7_8.  Google Scholar

[14] S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511546907.  Google Scholar
[15]

B. GordonW. Mills and L. Welch, Some new difference sets, Canadian Journal of Mathematics, 14 (1962), 614-625.  doi: 10.4153/CJM-1962-052-2.  Google Scholar

[16]

M. Hall, A survey of difference sets, Proceedings of the American Mathematical Society, 7 (1956), 975-986.  doi: 10.1090/S0002-9939-1956-0082502-7.  Google Scholar

[17]

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

[18]

T. Helleseth and P. V. Kumar, Sequences with low correlation, in Handbook of coding theory, Vol. I, II  Google Scholar

[19]

T. HellesethP. V. Kumar and H. Martinsen, A new family of ternary sequences with ideal two-level autocorrelation function, Des. Codes Cryptogr., 23 (2001), 157-166.  doi: 10.1023/A:1011208514883.  Google Scholar

[20]

J. R. HollonM. Rangaswamy and P. Setlur, New families of optimal high-energy ternary sequences having good correlation properties, Journal of Algebraic Combinatorics, 50 (2019), 1-38.  doi: 10.1007/s10801-018-0835-1.  Google Scholar

[21]

H. HuS. ShaoG. Gong and T. Helleseth, The proof of Lin's conjecture via the decimation-Hadamard transform, IEEE Transactions on Information Theory, 60 (2014), 5054-5064.  doi: 10.1109/TIT.2014.2327625.  Google Scholar

[22]

D. Jungnickel and A. Pott, Perfect and almost perfect sequences, Discrete Applied Mathematics, 95 (1999), 331-359.  doi: 10.1016/S0166-218X(99)00085-2.  Google Scholar

[23]

Y.-S. KimJ.-S. ChungJ.-S. No and H. Chung, On the autocorrelation distributions of Sidel'nikov sequences, IEEE Transactions on Information Theory, 51 (2005), 3303-3307.  doi: 10.1109/TIT.2005.853310.  Google Scholar

[24]

Y.-S. Kim, J.-W. Jang, S.-H. Kim and J.-S. No, New quaternary sequences with optimal autocorrelation, ISIT, (2009), 286–289. Google Scholar

[25]

Y.-S. KimJ.-W. JangS.-H. Kim and J.-S. No, New quaternary sequences with ideal autocorrelation constructed from Legendre sequences, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 96 (2013), 1872-1882.  doi: 10.1587/transfun.E96.A.1872.  Google Scholar

[26]

P. V. KumarR. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, Journal of Combinatorial Theory, Series A, 40 (1985), 90-107.  doi: 10.1016/0097-3165(85)90049-4.  Google Scholar

[27]

A. LempelM. Cohn and W. Eastman, A class of balanced binary sequences with optimal autocorrelation properties, IEEE Transactions on Information Theory, 23 (1977), 38-42.  doi: 10.1109/tit.1977.1055672.  Google Scholar

[28]

S. L. Ma and W. S. Ng, On non-existence of perfect and nearly perfect sequences, International Journal of Information and Coding Theory, 1 (2009), 15-38.  doi: 10.1504/IJICOT.2009.024045.  Google Scholar

[29]

S. L. Ma and B. Schmidt, On $(p^a, p, p^a, p^a-1)$-relative difference sets, Designs, Codes and Cryptography, 6 (1995), 57-71.  doi: 10.1007/BF01390771.  Google Scholar

[30]

A. Maschietti, Difference sets and hyperovals, Designs, Codes and Cryptography, 14 (1998), 89-98.  doi: 10.1023/A:1008264606494.  Google Scholar

[31]

J. Michel and Q. Wang, Some new balanced and almost balanced quaternary sequences with low autocorrelation, Cryptography and Communications, 11 (2019), 191-206.  doi: 10.1007/s12095-018-0281-x.  Google Scholar

[32]

J.-S. NoH. Chung and M.-S. Yun, Binary pseudorandom sequences of period $2^n-1$ with ideal autocorrelation generated by the polynomial $z^d+(z+ 1)^d$, IEEE Transactions on Information Theory, 44 (1998), 1278-1282.  doi: 10.1109/18.669400.  Google Scholar

[33]

B. Özden and O. Yayla, Cryptographic functions and bit-error-rate analysis with almost $ p $-ary sequences, International Journal of Information Security Science, 8.3 (2019), 44-52.  doi: 10.1007/s12095-020-00423-5.  Google Scholar

[34]

B. Özden and O. Yayla, Almost p-ary sequences, Cryptography and Communications, 12 (2020), 1057-1069.  doi: 10.1007/s12095-020-00423-5.  Google Scholar

[35]

R. E. Paley, On orthogonal matrices, Journal of Mathematics and Physics, 12 (1933), 311-320.  doi: 10.1002/sapm1933121311.  Google Scholar

[36]

A. Pott, Finite Geometry and Character Theory, Lecture Notes in Mathematics, 1601, Springer-Verlag, Berlin, 1995. doi: 10.1007/BFb0094449.  Google Scholar

[37]

K.-U. Schmidt, Quaternary constant-amplitude codes for multicode CDMA, IEEE Trans. Information Theory, 55 (2009), 1824-1832.  doi: 10.1109/TIT.2009.2013041.  Google Scholar

[38]

X. ShiX. ZhuX. Huang and Q. Yue, A family of $m$-ary $\sigma$-sequences with good autocorrelation, IEEE Communications Letters, 23 (2019), 1132-1135.  doi: 10.1109/LCOMM.2019.2915234.  Google Scholar

[39]

G. L. Sicuranza and A. Carini, Nonlinear system identification using quasi-perfect periodic sequences, Signal Processing, 120 (2016), 174-184.  doi: 10.1016/j.sigpro.2015.08.018.  Google Scholar

[40]

V. M. Sidel'nikov, Some k-valued pseudo-random sequences and nearly equidistant codes, Problemy Peredachi Informatsii, 5 (1969), 16-22.   Google Scholar

[41]

J. Singer, A theorem in finite projective geometry and some applications to number theory, Transactions of the American Mathematical Society, 43 (1938), 377-385.  doi: 10.1090/S0002-9947-1938-1501951-4.  Google Scholar

[42]

R. G. Stanton and D. Sprott, A family of difference sets, Canadian Journal of Mathematics, 10 (1958), 73-77.  doi: 10.4153/CJM-1958-008-5.  Google Scholar

[43]

T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, 2, Markham Publishing Co., Chicago, IL, 1967.  Google Scholar

[44]

X. Tang and C. Ding, New classes of balanced quaternary and almost balanced binary sequences with optimal autocorrelation value, IEEE Transactions on Information Theory, 56 (2010), 6398-6405.  doi: 10.1109/TIT.2010.2081170.  Google Scholar

[45]

X. Tang and G. Gong, New constructions of binary sequences with optimal autocorrelation value/magnitude, IEEE Transactions on Information Theory, 56 (2010), 1278-1286.  doi: 10.1109/TIT.2009.2039159.  Google Scholar

[46]

X. Tang and J. Lindner, Almost quadriphase sequence with ideal autocorrelation property, IEEE Signal Processing Letters, 16 (2008), 38-40.   Google Scholar

[47]

A. Tirkel and T. Hall, New quasi-perfect and perfect sequences of roots of unity and zero, in International Conference on Sequences and Their Applications Google Scholar

[48]

Q. Wang, W. Kong, Y. Yan, C. Wu and M. Yang, Autocorrelation of a class of quaternary sequences of period $2 p^m$, preprint, arXiv: 2002.00375. Google Scholar

[49]

O. Yayla, Nearly perfect sequences with arbitrary out-of-phase autocorrelation, Advances in Mathematics of Communications, 10 (2016), 401-411.  doi: 10.3934/amc.2016014.  Google Scholar

[50]

N. Y. Yu and G. Gong, New binary sequences with optimal autocorrelation magnitude, IEEE Transactions on Information Theory, 54 (2008), 4771-4779.  doi: 10.1109/TIT.2008.928999.  Google Scholar

show all references

References:
[1]

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

[2]

K. T. ArasuC. DingT. HellesethP. V. Kumar and H. M. Martinsen, Almost difference sets and their sequences with optimal autocorrelation, IEEE Transactions on Information Theory, 47 (2001), 2934-2943.  doi: 10.1109/18.959271.  Google Scholar

[3] T. BethD. Jungnickel and H. Lenz, Design Theory: Volume 1, Cambridge University Press, 1999.   Google Scholar
[4]

Y. Cai and C. Ding, Binary sequences with optimal autocorrelation, Theoretical Computer Science, 410 (2009), 2316-2322.  doi: 10.1016/j.tcs.2009.02.021.  Google Scholar

[5]

A. Çeșmelioǧlu and O. Olmez, Graphs of vectorial plateaued functions as difference sets, Finite Fields and Their Applications, 71 (2021), 101795. doi: 10.1016/j.ffa.2020.101795.  Google Scholar

[6]

Y. M. Chee, Y. Tan and Y. Zhou, Almost p-ary perfect sequences, in International Conference on Sequences and Their Applications doi: 10.1007/978-3-642-15874-2_34.  Google Scholar

[7]

I. Chih-Lin and R. D. Gitlin, Multi-code CDMA wireless personal communications networks, Proceedings IEEE International Conference on Communications ICC'95, 2 (1995), 1060-1064.  doi: 10.1109/ICC.1995.524263.  Google Scholar

[8] C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, CRC press, 2006.  doi: 10.1201/9781420049954.  Google Scholar
[9]

L. E. Dickson, Cyclotomy, higher congruences, and Waring's problem, American Journal of Mathematics, 57 (1935), 391-424.  doi: 10.2307/2371217.  Google Scholar

[10]

J. F. Dillon and H. Dobbertin, New cyclic difference sets with singer parameters, Finite Fields and Their Applications, 10 (2004), 342-389.  doi: 10.1016/j.ffa.2003.09.003.  Google Scholar

[11]

C. DingT. Helleseth and K. Y. Lam, Several classes of binary sequences with three-level autocorrelation, IEEE Transactions on Information Theory, 45 (1999), 2606-2612.  doi: 10.1109/18.796414.  Google Scholar

[12]

C. DingT. Helleseth and H. Martinsen, New families of binary sequences with optimal three-level autocorrelation, IEEE Transactions on Information Theory, 47 (2001), 428-433.  doi: 10.1109/18.904555.  Google Scholar

[13]

V. E. Gantmakher and M. V. Zaleshin, Almost six-phase sequences with perfect periodic autocorrelation function, in International Conference on Sequences and Their Applications doi: 10.1007/978-3-319-12325-7_8.  Google Scholar

[14] S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511546907.  Google Scholar
[15]

B. GordonW. Mills and L. Welch, Some new difference sets, Canadian Journal of Mathematics, 14 (1962), 614-625.  doi: 10.4153/CJM-1962-052-2.  Google Scholar

[16]

M. Hall, A survey of difference sets, Proceedings of the American Mathematical Society, 7 (1956), 975-986.  doi: 10.1090/S0002-9939-1956-0082502-7.  Google Scholar

[17]

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

[18]

T. Helleseth and P. V. Kumar, Sequences with low correlation, in Handbook of coding theory, Vol. I, II  Google Scholar

[19]

T. HellesethP. V. Kumar and H. Martinsen, A new family of ternary sequences with ideal two-level autocorrelation function, Des. Codes Cryptogr., 23 (2001), 157-166.  doi: 10.1023/A:1011208514883.  Google Scholar

[20]

J. R. HollonM. Rangaswamy and P. Setlur, New families of optimal high-energy ternary sequences having good correlation properties, Journal of Algebraic Combinatorics, 50 (2019), 1-38.  doi: 10.1007/s10801-018-0835-1.  Google Scholar

[21]

H. HuS. ShaoG. Gong and T. Helleseth, The proof of Lin's conjecture via the decimation-Hadamard transform, IEEE Transactions on Information Theory, 60 (2014), 5054-5064.  doi: 10.1109/TIT.2014.2327625.  Google Scholar

[22]

D. Jungnickel and A. Pott, Perfect and almost perfect sequences, Discrete Applied Mathematics, 95 (1999), 331-359.  doi: 10.1016/S0166-218X(99)00085-2.  Google Scholar

[23]

Y.-S. KimJ.-S. ChungJ.-S. No and H. Chung, On the autocorrelation distributions of Sidel'nikov sequences, IEEE Transactions on Information Theory, 51 (2005), 3303-3307.  doi: 10.1109/TIT.2005.853310.  Google Scholar

[24]

Y.-S. Kim, J.-W. Jang, S.-H. Kim and J.-S. No, New quaternary sequences with optimal autocorrelation, ISIT, (2009), 286–289. Google Scholar

[25]

Y.-S. KimJ.-W. JangS.-H. Kim and J.-S. No, New quaternary sequences with ideal autocorrelation constructed from Legendre sequences, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 96 (2013), 1872-1882.  doi: 10.1587/transfun.E96.A.1872.  Google Scholar

[26]

P. V. KumarR. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, Journal of Combinatorial Theory, Series A, 40 (1985), 90-107.  doi: 10.1016/0097-3165(85)90049-4.  Google Scholar

[27]

A. LempelM. Cohn and W. Eastman, A class of balanced binary sequences with optimal autocorrelation properties, IEEE Transactions on Information Theory, 23 (1977), 38-42.  doi: 10.1109/tit.1977.1055672.  Google Scholar

[28]

S. L. Ma and W. S. Ng, On non-existence of perfect and nearly perfect sequences, International Journal of Information and Coding Theory, 1 (2009), 15-38.  doi: 10.1504/IJICOT.2009.024045.  Google Scholar

[29]

S. L. Ma and B. Schmidt, On $(p^a, p, p^a, p^a-1)$-relative difference sets, Designs, Codes and Cryptography, 6 (1995), 57-71.  doi: 10.1007/BF01390771.  Google Scholar

[30]

A. Maschietti, Difference sets and hyperovals, Designs, Codes and Cryptography, 14 (1998), 89-98.  doi: 10.1023/A:1008264606494.  Google Scholar

[31]

J. Michel and Q. Wang, Some new balanced and almost balanced quaternary sequences with low autocorrelation, Cryptography and Communications, 11 (2019), 191-206.  doi: 10.1007/s12095-018-0281-x.  Google Scholar

[32]

J.-S. NoH. Chung and M.-S. Yun, Binary pseudorandom sequences of period $2^n-1$ with ideal autocorrelation generated by the polynomial $z^d+(z+ 1)^d$, IEEE Transactions on Information Theory, 44 (1998), 1278-1282.  doi: 10.1109/18.669400.  Google Scholar

[33]

B. Özden and O. Yayla, Cryptographic functions and bit-error-rate analysis with almost $ p $-ary sequences, International Journal of Information Security Science, 8.3 (2019), 44-52.  doi: 10.1007/s12095-020-00423-5.  Google Scholar

[34]

B. Özden and O. Yayla, Almost p-ary sequences, Cryptography and Communications, 12 (2020), 1057-1069.  doi: 10.1007/s12095-020-00423-5.  Google Scholar

[35]

R. E. Paley, On orthogonal matrices, Journal of Mathematics and Physics, 12 (1933), 311-320.  doi: 10.1002/sapm1933121311.  Google Scholar

[36]

A. Pott, Finite Geometry and Character Theory, Lecture Notes in Mathematics, 1601, Springer-Verlag, Berlin, 1995. doi: 10.1007/BFb0094449.  Google Scholar

[37]

K.-U. Schmidt, Quaternary constant-amplitude codes for multicode CDMA, IEEE Trans. Information Theory, 55 (2009), 1824-1832.  doi: 10.1109/TIT.2009.2013041.  Google Scholar

[38]

X. ShiX. ZhuX. Huang and Q. Yue, A family of $m$-ary $\sigma$-sequences with good autocorrelation, IEEE Communications Letters, 23 (2019), 1132-1135.  doi: 10.1109/LCOMM.2019.2915234.  Google Scholar

[39]

G. L. Sicuranza and A. Carini, Nonlinear system identification using quasi-perfect periodic sequences, Signal Processing, 120 (2016), 174-184.  doi: 10.1016/j.sigpro.2015.08.018.  Google Scholar

[40]

V. M. Sidel'nikov, Some k-valued pseudo-random sequences and nearly equidistant codes, Problemy Peredachi Informatsii, 5 (1969), 16-22.   Google Scholar

[41]

J. Singer, A theorem in finite projective geometry and some applications to number theory, Transactions of the American Mathematical Society, 43 (1938), 377-385.  doi: 10.1090/S0002-9947-1938-1501951-4.  Google Scholar

[42]

R. G. Stanton and D. Sprott, A family of difference sets, Canadian Journal of Mathematics, 10 (1958), 73-77.  doi: 10.4153/CJM-1958-008-5.  Google Scholar

[43]

T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, 2, Markham Publishing Co., Chicago, IL, 1967.  Google Scholar

[44]

X. Tang and C. Ding, New classes of balanced quaternary and almost balanced binary sequences with optimal autocorrelation value, IEEE Transactions on Information Theory, 56 (2010), 6398-6405.  doi: 10.1109/TIT.2010.2081170.  Google Scholar

[45]

X. Tang and G. Gong, New constructions of binary sequences with optimal autocorrelation value/magnitude, IEEE Transactions on Information Theory, 56 (2010), 1278-1286.  doi: 10.1109/TIT.2009.2039159.  Google Scholar

[46]

X. Tang and J. Lindner, Almost quadriphase sequence with ideal autocorrelation property, IEEE Signal Processing Letters, 16 (2008), 38-40.   Google Scholar

[47]

A. Tirkel and T. Hall, New quasi-perfect and perfect sequences of roots of unity and zero, in International Conference on Sequences and Their Applications Google Scholar

[48]

Q. Wang, W. Kong, Y. Yan, C. Wu and M. Yang, Autocorrelation of a class of quaternary sequences of period $2 p^m$, preprint, arXiv: 2002.00375. Google Scholar

[49]

O. Yayla, Nearly perfect sequences with arbitrary out-of-phase autocorrelation, Advances in Mathematics of Communications, 10 (2016), 401-411.  doi: 10.3934/amc.2016014.  Google Scholar

[50]

N. Y. Yu and G. Gong, New binary sequences with optimal autocorrelation magnitude, IEEE Transactions on Information Theory, 54 (2008), 4771-4779.  doi: 10.1109/TIT.2008.928999.  Google Scholar

Table 1.  Orbits of $ G = {\mathbb Z}_{10} \times {\mathbb Z}_3 $ under $ x \rightarrow 19x $
orbits of length 1
(0, 0)}, {(0, 1)}, {(0, 2)}, {(5, 0)}, {(5, 1)}, {(5, 2)
orbits of length 2
(4, 0), (6, 0)}, {(1, 1), (9, 1)}, {(7, 0), (3, 0)}, {(8, 2), (2, 2)},
(9, 0), (1, 0)} {(7, 2), (3, 2)}, {(7, 1), (3, 1)}, {(1, 2), (9, 2)},
(8, 1), (2, 1)}, {(6, 2), (4, 2)}, {(6, 1), (4, 1)}, {(2, 0), (8, 0)
orbits of length 1
(0, 0)}, {(0, 1)}, {(0, 2)}, {(5, 0)}, {(5, 1)}, {(5, 2)
orbits of length 2
(4, 0), (6, 0)}, {(1, 1), (9, 1)}, {(7, 0), (3, 0)}, {(8, 2), (2, 2)},
(9, 0), (1, 0)} {(7, 2), (3, 2)}, {(7, 1), (3, 1)}, {(1, 2), (9, 2)},
(8, 1), (2, 1)}, {(6, 2), (4, 2)}, {(6, 1), (4, 1)}, {(2, 0), (8, 0)
Table 2.  Sequences, their autocorrelation and alphabet
Construction Out-of-phase autocorrelation Alphabet
[29], [36] 0 $ p $-ary
[17], [18] -1 $ p $-ary
[3], [10], [15], [16], [30], [32], [35], [41], [42], [43], [44] -1 binary
[12], [27], [40], [44] $ \pm 2 $ binary
[2], [40], [27], [44] $ (0,-4) $ binary
[45], [50] $ (0,\pm 4) $ binary
[4], [11], [44] $ (1,-3) $ binary
[25] $ (2p,-2) $ or $ (\pm 2p, \pm 2) $ binary
[1], [19], [21] -1 ternary
[23] $ (0,-3,3\zeta_3,3\zeta_3^2) $ ternary
$ (0,\pm 2i,-4,-2,-2\pm 2i) $ or $ (0,\pm 2i,\pm2,-2\pm 2i) $ quaternary
[24] $ (-2,\pm 2i) $ quaternary
[25], [44] $ (0,-2) $ quaternary
[31] $ (-1,\pm 3) $ quaternary
[46] $ (-1,\pm(1+2i)) $ or $ (\pm 1,-3) $ quaternary
[48] $ (\frac{p^{n-1}(p-7)}{2},\frac{p^{n-1}(p-3)}{2},p^n) $ quaternary
[6] $ 0 $ $ p $-ary with one zero
$ -1 $ $ p $-ary with one zero
[38] $ -1 $ $ m $-ary with one zero
[13] $ (0,3q^{n-1}) $ $ 6 $-ary with one zero
[47] $ (0,p^{(k-1)n}) $ $ p^{kn} $-ary with zeros
$ (0,p^{(k-1)n}) $ $ \frac{p^{n-1}}{\gcd(t,p^{n-1})} $-ary with zeros
$ 0 $ $ m $-ary with one zero
Theorem 3.7 $ \frac{q-3}{2} $ quaternary with one zero
Proposition 7 $ (-1,0) $ $ m $-ary with $ \frac{q}{2}+1 $ zeros
Construction Out-of-phase autocorrelation Alphabet
[29], [36] 0 $ p $-ary
[17], [18] -1 $ p $-ary
[3], [10], [15], [16], [30], [32], [35], [41], [42], [43], [44] -1 binary
[12], [27], [40], [44] $ \pm 2 $ binary
[2], [40], [27], [44] $ (0,-4) $ binary
[45], [50] $ (0,\pm 4) $ binary
[4], [11], [44] $ (1,-3) $ binary
[25] $ (2p,-2) $ or $ (\pm 2p, \pm 2) $ binary
[1], [19], [21] -1 ternary
[23] $ (0,-3,3\zeta_3,3\zeta_3^2) $ ternary
$ (0,\pm 2i,-4,-2,-2\pm 2i) $ or $ (0,\pm 2i,\pm2,-2\pm 2i) $ quaternary
[24] $ (-2,\pm 2i) $ quaternary
[25], [44] $ (0,-2) $ quaternary
[31] $ (-1,\pm 3) $ quaternary
[46] $ (-1,\pm(1+2i)) $ or $ (\pm 1,-3) $ quaternary
[48] $ (\frac{p^{n-1}(p-7)}{2},\frac{p^{n-1}(p-3)}{2},p^n) $ quaternary
[6] $ 0 $ $ p $-ary with one zero
$ -1 $ $ p $-ary with one zero
[38] $ -1 $ $ m $-ary with one zero
[13] $ (0,3q^{n-1}) $ $ 6 $-ary with one zero
[47] $ (0,p^{(k-1)n}) $ $ p^{kn} $-ary with zeros
$ (0,p^{(k-1)n}) $ $ \frac{p^{n-1}}{\gcd(t,p^{n-1})} $-ary with zeros
$ 0 $ $ m $-ary with one zero
Theorem 3.7 $ \frac{q-3}{2} $ quaternary with one zero
Proposition 7 $ (-1,0) $ $ m $-ary with $ \frac{q}{2}+1 $ zeros
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