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Double-circulant and bordered-double-circulant constructions for self-dual codes over $R_2$
Good random matrices over finite fields
1. | Zhengyuan Xiaoqu 10-2-101, Fengtan Road, Hangzhou 310011, China |
2. | Department of Information Science and Electronics Engineering, Zhejiang University, 38 Zheda Road, Hangzhou 310027, China |
References:
[1] |
G. E. Andrews, "The Theory of Partitions,'' Cambridge University Press, 1998. |
[2] |
S. Ball, The polynomial method in Galois geometries,, in, (): 103.
|
[3] |
A. Barg and G. D. Forney, Jr., Random codes: Minimum distances and error exponents, IEEE Trans. Inform. Theory, 48 (2002), 2568-2573.
doi: 10.1109/TIT.2002.800480. |
[4] |
J. D. Beule and L. Storme (eds.), "Current Research Topics in Galois Geometry,'' Nova Science Publishers, 2011. |
[5] |
A. Blokhuis, P. Sziklai and T. Szőnyi, Blocking sets in projective spaces,, in, (): 61.
|
[6] |
B. Bose and T. R. N. Rao, Separating and completely separating systems and linear codes, IEEE Trans. Comput., 29 (1980), 665-668.
doi: 10.1109/TC.1980.1675640. |
[7] |
A. E. Brouwer and A. Schrijver, The blocking number of an affine space, J. Combin. Theory Ser. A, 24 (1978), 251-253.
doi: 10.1016/0097-3165(78)90013-4. |
[8] |
F. de Clerck and H. van Maldeghem, Some classes of rank two geometries, in "Handbook of Incidence Geometry--Buildings and Foundations'' (ed. F. Buekenhout), Elsevier Science Publ., (1995), 433-475. |
[9] |
G. Cohen and G. Zémor, Intersecting codes and independent families, IEEE Trans. Inform. Theory, 40 (1994), 1872-1881.
doi: 10.1109/18.340462. |
[10] |
G. Cohen and G. Zémor, Copyright protection for digital data, IEEE Commun. Lett., 4 (2000), 158-160.
doi: 10.1109/4234.846497. |
[11] |
I. Csiszár, Linear codes for sources and source networks: error exponents, universal coding, IEEE Trans. Inform. Theory, 28 (1982), 585-592.
doi: 10.1109/TIT.1982.1056524. |
[12] |
P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241.
doi: 10.1016/0097-3165(78)90015-8. |
[13] | |
[14] |
A. B. Evans, "Orthomorphism Graphs of Groups,'' Springer-Verlag, 1992. |
[15] |
E.M. Gabidulin, Theory of codes with maximum rank distance, Probl. Inf. Transm., 21 (1985), 1-12. |
[16] |
R. G. Gallager, "Low-Density Parity-Check Codes,'' MIT Press, Cambridge, MA, 1963. |
[17] |
M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams' equivalence theorem, J. Combin. Theory Ser. A, 92 (2000), 17-28.
doi: 10.1006/jcta.1999.3033. |
[18] |
M. Hall, Jr., "Combinatorial Theory,'' 2nd edition, John Wiley & Sons, 1986. |
[19] |
W. Heise and T. Honold, Homogeneous and egalitarian weights on finite rings, in "Proceedings of the Seventh International Workshop on Algebraic and Combinatorial Coding Theory (ACCT-2000),'' Bansko, Bulgaria, (2000), 183-188. |
[20] |
J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2nd edition, Oxford University Press, New York, 1998. |
[21] |
J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces, J. Statist. Plann. Inference, 72 (1998), 355-380.
doi: 10.1016/S0378-3758(98)00043-3. |
[22] |
J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces: update 2001, in "Finite Geometries,'' Kluwer Acad. Publ., Dordrecht, (2001), 201-246.
doi: 10.1007/978-1-4613-0283-4_13. |
[23] |
T. Honold and A. A. Nechaev, Weighted modules and representations of codes, Probl. Inform. Transm., 35 (1999), 205-223. |
[24] |
R. E. Jamison, Covering finite fields with cosets of subspaces, J. Combin. Theory Ser. A, 22 (1977), 253-266.
doi: 10.1016/0097-3165(77)90001-2. |
[25] |
D. J. Kleitman and J. Spencer, Families of k-independent sets, Discrete Math., 6 (1973), 255-262.
doi: 10.1016/0012-365X(73)90098-8. |
[26] |
J. Körner, On the extremal combinatorics of the Hamming space, J. Combin. Theory Ser. A, 71 (1995), 112-126.
doi: 10.1016/0097-3165(95)90019-5. |
[27] |
T.-Y. Lam, "A First Course in Noncommutative Rings,'' Springer-Verlag, 1991.
doi: 10.1007/978-1-4684-0406-7. |
[28] |
I. Landjev and L. Storme, Galois geometries and coding theory,, in, (): 185.
|
[29] |
H. Niederreiter and K. H. Robinson, Complete mappings of finite fields, J. Aust. Math. Soc. Ser. A, 33 (1982), 197-212.
doi: 10.1017/S1446788700018346. |
[30] |
D. R. Pradhan and S. M. Peddy, Techniques to construct $(2, 1)$ separating systems from linear error-correcting codes, IEEE Trans. Comput., 25 (1976), 945-949.
doi: 10.1109/TC.1976.1674720. |
[31] |
R. M. Roth, Maximum-rank array codes and their application to crisscross error correction, IEEE Trans. Inform. Theory, 37 (1991), 328-336.
doi: 10.1109/18.75248. |
[32] |
Z.-X. Wan, "Geometry of Matrices,'' World Scientific, 1996.
doi: 10.1142/9789812830234. |
[33] |
S. Yang, Y. Chen and P. Qiu, Linear-codes-based lossless joint source-channel coding for multiple-access channels, IEEE Trans. Inform. Theory, 55 (2009), 1468-1486.
doi: 10.1109/TIT.2009.2013009. |
[34] |
S. Yang, T. Honold, Y. Chen, Z. Zhang and P. Qiu, Constructing linear codes with good spectra,, preprint, ().
|
[35] |
Y. Yuan, Y. Tong and H. Zhang, Complete mapping polynomials over finite field $\mathbb F_{16}$, in "Arithmetic of Finite Fields,'' Springer-Verlag, Berlin, (2007), 147-158.
doi: 10.1007/978-3-540-73074-3_12. |
show all references
References:
[1] |
G. E. Andrews, "The Theory of Partitions,'' Cambridge University Press, 1998. |
[2] |
S. Ball, The polynomial method in Galois geometries,, in, (): 103.
|
[3] |
A. Barg and G. D. Forney, Jr., Random codes: Minimum distances and error exponents, IEEE Trans. Inform. Theory, 48 (2002), 2568-2573.
doi: 10.1109/TIT.2002.800480. |
[4] |
J. D. Beule and L. Storme (eds.), "Current Research Topics in Galois Geometry,'' Nova Science Publishers, 2011. |
[5] |
A. Blokhuis, P. Sziklai and T. Szőnyi, Blocking sets in projective spaces,, in, (): 61.
|
[6] |
B. Bose and T. R. N. Rao, Separating and completely separating systems and linear codes, IEEE Trans. Comput., 29 (1980), 665-668.
doi: 10.1109/TC.1980.1675640. |
[7] |
A. E. Brouwer and A. Schrijver, The blocking number of an affine space, J. Combin. Theory Ser. A, 24 (1978), 251-253.
doi: 10.1016/0097-3165(78)90013-4. |
[8] |
F. de Clerck and H. van Maldeghem, Some classes of rank two geometries, in "Handbook of Incidence Geometry--Buildings and Foundations'' (ed. F. Buekenhout), Elsevier Science Publ., (1995), 433-475. |
[9] |
G. Cohen and G. Zémor, Intersecting codes and independent families, IEEE Trans. Inform. Theory, 40 (1994), 1872-1881.
doi: 10.1109/18.340462. |
[10] |
G. Cohen and G. Zémor, Copyright protection for digital data, IEEE Commun. Lett., 4 (2000), 158-160.
doi: 10.1109/4234.846497. |
[11] |
I. Csiszár, Linear codes for sources and source networks: error exponents, universal coding, IEEE Trans. Inform. Theory, 28 (1982), 585-592.
doi: 10.1109/TIT.1982.1056524. |
[12] |
P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241.
doi: 10.1016/0097-3165(78)90015-8. |
[13] | |
[14] |
A. B. Evans, "Orthomorphism Graphs of Groups,'' Springer-Verlag, 1992. |
[15] |
E.M. Gabidulin, Theory of codes with maximum rank distance, Probl. Inf. Transm., 21 (1985), 1-12. |
[16] |
R. G. Gallager, "Low-Density Parity-Check Codes,'' MIT Press, Cambridge, MA, 1963. |
[17] |
M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams' equivalence theorem, J. Combin. Theory Ser. A, 92 (2000), 17-28.
doi: 10.1006/jcta.1999.3033. |
[18] |
M. Hall, Jr., "Combinatorial Theory,'' 2nd edition, John Wiley & Sons, 1986. |
[19] |
W. Heise and T. Honold, Homogeneous and egalitarian weights on finite rings, in "Proceedings of the Seventh International Workshop on Algebraic and Combinatorial Coding Theory (ACCT-2000),'' Bansko, Bulgaria, (2000), 183-188. |
[20] |
J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2nd edition, Oxford University Press, New York, 1998. |
[21] |
J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces, J. Statist. Plann. Inference, 72 (1998), 355-380.
doi: 10.1016/S0378-3758(98)00043-3. |
[22] |
J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces: update 2001, in "Finite Geometries,'' Kluwer Acad. Publ., Dordrecht, (2001), 201-246.
doi: 10.1007/978-1-4613-0283-4_13. |
[23] |
T. Honold and A. A. Nechaev, Weighted modules and representations of codes, Probl. Inform. Transm., 35 (1999), 205-223. |
[24] |
R. E. Jamison, Covering finite fields with cosets of subspaces, J. Combin. Theory Ser. A, 22 (1977), 253-266.
doi: 10.1016/0097-3165(77)90001-2. |
[25] |
D. J. Kleitman and J. Spencer, Families of k-independent sets, Discrete Math., 6 (1973), 255-262.
doi: 10.1016/0012-365X(73)90098-8. |
[26] |
J. Körner, On the extremal combinatorics of the Hamming space, J. Combin. Theory Ser. A, 71 (1995), 112-126.
doi: 10.1016/0097-3165(95)90019-5. |
[27] |
T.-Y. Lam, "A First Course in Noncommutative Rings,'' Springer-Verlag, 1991.
doi: 10.1007/978-1-4684-0406-7. |
[28] |
I. Landjev and L. Storme, Galois geometries and coding theory,, in, (): 185.
|
[29] |
H. Niederreiter and K. H. Robinson, Complete mappings of finite fields, J. Aust. Math. Soc. Ser. A, 33 (1982), 197-212.
doi: 10.1017/S1446788700018346. |
[30] |
D. R. Pradhan and S. M. Peddy, Techniques to construct $(2, 1)$ separating systems from linear error-correcting codes, IEEE Trans. Comput., 25 (1976), 945-949.
doi: 10.1109/TC.1976.1674720. |
[31] |
R. M. Roth, Maximum-rank array codes and their application to crisscross error correction, IEEE Trans. Inform. Theory, 37 (1991), 328-336.
doi: 10.1109/18.75248. |
[32] |
Z.-X. Wan, "Geometry of Matrices,'' World Scientific, 1996.
doi: 10.1142/9789812830234. |
[33] |
S. Yang, Y. Chen and P. Qiu, Linear-codes-based lossless joint source-channel coding for multiple-access channels, IEEE Trans. Inform. Theory, 55 (2009), 1468-1486.
doi: 10.1109/TIT.2009.2013009. |
[34] |
S. Yang, T. Honold, Y. Chen, Z. Zhang and P. Qiu, Constructing linear codes with good spectra,, preprint, ().
|
[35] |
Y. Yuan, Y. Tong and H. Zhang, Complete mapping polynomials over finite field $\mathbb F_{16}$, in "Arithmetic of Finite Fields,'' Springer-Verlag, Berlin, (2007), 147-158.
doi: 10.1007/978-3-540-73074-3_12. |
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