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On optimal ternary linear codes of dimension 6
New nearly optimal codebooks from relative difference sets
1. | School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China |
2. | Provincial Key Lab of Information Coding and Transmission, Southwest Jiaotong University, Chengdu, 610031, China |
References:
[1] |
K. T. Arasu, J. F. Dillon, D. Jungnickel and A. Pott, The solution of the Waterloo problem, J. Combin. Theory Ser. A, 71 (1995), 316-331.
doi: 10.1016/0097-3165(95)90006-3. |
[2] |
K. T. Arasu, J. F. Dillon, K. H. Leung and S. L. Ma, Cyclic relative difference sets with classical parameters, J. Combin. Theory Ser. A, 94 (2001), 118-126.
doi: 10.1006/jcta.2000.3137. |
[3] |
R. C. Bose, An affine analog of Singer's theorem, J. Indian Math. Soc., 6 (1942), 1-15. |
[4] |
D. Chandler and Q. Xiang, Cyclic relative difference sets and their p-ranks, Des. Codes Cryptogr., 30 (2003), 325-343.
doi: 10.1023/A:1025750228679. |
[5] |
J. H. Conway, R. H. Harding and N. J. A. Sloane, Packing lines, planes, etc.: Packings in grassmannian spaces, Exp. Math., 5 (1996), 139-159. |
[6] |
C. Ding, Complex codebooks from combinatorial designs, IEEE Trans. Inform. Theory, 52 (2006), 4229-4235.
doi: 10.1109/TIT.2006.880058. |
[7] |
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. |
[8] |
C. Ding and T. Feng, Codebooks from almost difference sets, Des. Codes Cryptogr., 46 (2008), 113-126.
doi: 10.1007/s10623-007-9140-z. |
[9] |
S.-H. Kim, J.-S. No, H.-C. Chung and T. Helleseth, New cyclic relative difference sets constructed from $d$-homogeneous functions with difference-balanced property, IEEE Trans. Inform. Theory, 51 (2005), 1155-1163. |
[10] |
P. V. Kumar, On the existence of square dot-matrix patterns having a specific three-valued periodic-correlation function, IEEE Trans. Inform. Theory, 34 (1988), 271-277.
doi: 10.1109/18.2635. |
[11] |
K. H. Leung and S. L. Ma, Constructions of relative difference sets with classical parameters and circulant weighing matrices, J. Combin. Theory Ser. A, 99 (2002), 111-127 .
doi: 10.1006/jcta.2002.3262. |
[12] |
R. Lidl and H. Niederreiter, "Finite Fields,'' Addison-Wesley, Reading, MA, 1983. |
[13] |
S. L. Ma and A. Pott, Relative difference sets, planar functions, and generalized Hadamard matrices, J. Algebra, 175 (1995), 505-525.
doi: 10.1006/jabr.1995.1198. |
[14] |
J. L. Massey and T. Mittlelholzer, Welch's bound and sequence sets for code-division multiple-access systems, in "Sequences II: Methods in Communication, Security and Computer Science,'' Springer, Heidelberg, New York, (1993), 63-78. |
[15] |
A. Pott, "Finite Geometry and Character Theory,'' Springer, 1995. |
[16] |
A. Pott, A survey on relative difference sets, in "Groups, Difference Sets, and the Monster'' (eds. K.T. Arasu, et al.), Walter de Gruyter, (1996), 195-232. |
[17] |
D. Sarwate, Meeting the Welch bound with equality, in "Proc. of SETA'98,'' Springer-Verlag, Berlin, Heidelberg, (1999), 79-102. |
[18] |
T. Strohmer and R. W. Heath Jr., Grassmannian frames with applications to coding and communication, Appl. Comput. Harmonic Anal., 14 (2003), 257-275.
doi: 10.1016/S1063-5203(03)00023-X. |
[19] |
L. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Theory, IT-20 (1974), 397-399.
doi: 10.1109/TIT.1974.1055219. |
[20] |
P. Xia, S. Zhou and G. B. Giannakis, Achieving the Welch bound with difference sets, IEEE Trans. Inform. Theory, 51 (2005), 1900-1907.
doi: 10.1109/TIT.1974.1055219. |
show all references
References:
[1] |
K. T. Arasu, J. F. Dillon, D. Jungnickel and A. Pott, The solution of the Waterloo problem, J. Combin. Theory Ser. A, 71 (1995), 316-331.
doi: 10.1016/0097-3165(95)90006-3. |
[2] |
K. T. Arasu, J. F. Dillon, K. H. Leung and S. L. Ma, Cyclic relative difference sets with classical parameters, J. Combin. Theory Ser. A, 94 (2001), 118-126.
doi: 10.1006/jcta.2000.3137. |
[3] |
R. C. Bose, An affine analog of Singer's theorem, J. Indian Math. Soc., 6 (1942), 1-15. |
[4] |
D. Chandler and Q. Xiang, Cyclic relative difference sets and their p-ranks, Des. Codes Cryptogr., 30 (2003), 325-343.
doi: 10.1023/A:1025750228679. |
[5] |
J. H. Conway, R. H. Harding and N. J. A. Sloane, Packing lines, planes, etc.: Packings in grassmannian spaces, Exp. Math., 5 (1996), 139-159. |
[6] |
C. Ding, Complex codebooks from combinatorial designs, IEEE Trans. Inform. Theory, 52 (2006), 4229-4235.
doi: 10.1109/TIT.2006.880058. |
[7] |
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. |
[8] |
C. Ding and T. Feng, Codebooks from almost difference sets, Des. Codes Cryptogr., 46 (2008), 113-126.
doi: 10.1007/s10623-007-9140-z. |
[9] |
S.-H. Kim, J.-S. No, H.-C. Chung and T. Helleseth, New cyclic relative difference sets constructed from $d$-homogeneous functions with difference-balanced property, IEEE Trans. Inform. Theory, 51 (2005), 1155-1163. |
[10] |
P. V. Kumar, On the existence of square dot-matrix patterns having a specific three-valued periodic-correlation function, IEEE Trans. Inform. Theory, 34 (1988), 271-277.
doi: 10.1109/18.2635. |
[11] |
K. H. Leung and S. L. Ma, Constructions of relative difference sets with classical parameters and circulant weighing matrices, J. Combin. Theory Ser. A, 99 (2002), 111-127 .
doi: 10.1006/jcta.2002.3262. |
[12] |
R. Lidl and H. Niederreiter, "Finite Fields,'' Addison-Wesley, Reading, MA, 1983. |
[13] |
S. L. Ma and A. Pott, Relative difference sets, planar functions, and generalized Hadamard matrices, J. Algebra, 175 (1995), 505-525.
doi: 10.1006/jabr.1995.1198. |
[14] |
J. L. Massey and T. Mittlelholzer, Welch's bound and sequence sets for code-division multiple-access systems, in "Sequences II: Methods in Communication, Security and Computer Science,'' Springer, Heidelberg, New York, (1993), 63-78. |
[15] |
A. Pott, "Finite Geometry and Character Theory,'' Springer, 1995. |
[16] |
A. Pott, A survey on relative difference sets, in "Groups, Difference Sets, and the Monster'' (eds. K.T. Arasu, et al.), Walter de Gruyter, (1996), 195-232. |
[17] |
D. Sarwate, Meeting the Welch bound with equality, in "Proc. of SETA'98,'' Springer-Verlag, Berlin, Heidelberg, (1999), 79-102. |
[18] |
T. Strohmer and R. W. Heath Jr., Grassmannian frames with applications to coding and communication, Appl. Comput. Harmonic Anal., 14 (2003), 257-275.
doi: 10.1016/S1063-5203(03)00023-X. |
[19] |
L. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Theory, IT-20 (1974), 397-399.
doi: 10.1109/TIT.1974.1055219. |
[20] |
P. Xia, S. Zhou and G. B. Giannakis, Achieving the Welch bound with difference sets, IEEE Trans. Inform. Theory, 51 (2005), 1900-1907.
doi: 10.1109/TIT.1974.1055219. |
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