-
Previous Article
Probability estimates for reachability of linear systems defined over finite fields
- AMC Home
- This Issue
-
Next Article
On extendability of additive code isometries
Yet another variation on minimal linear codes
1. | Télécom ParisTech, UMR 5141, CNRS, 46 rue Barrault, 75634 Paris Cedex 13, France, France |
2. | University of Paris XIII and Paris VIII, Télécom ParisTech, LAGA, UMR 7539, CNRS, Sorbonne Paris Cité, France |
References:
[1] |
N. Alon, G. Cohen, M. Krivilevitch and S. Litsyn, Generalized hashing and applications, JCT-A, 104 (2003), 207-215.
doi: 10.1016/j.jcta.2003.08.001. |
[2] |
A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inf. Theory, 44 (1998), 2010-2017.
doi: 10.1109/18.705584. |
[3] |
A. Ashikhmin, A. Barg, G. Cohen and L. Huguet, Variations on minimal codewords in linear codes, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer, 1995, 96-105.
doi: 10.1007/3-540-60114-7_7. |
[4] |
A. Bassa, P. Beelen, A. Garcia and H. Stichtenoth, Towers of function fields over non-prime finite fields, Moscow Math. J., 15 (2015), 1-29. |
[5] |
G. Brassard, C. Crépeau and M. Santha, Oblivious transfers and intersecting codes, IEEE Trans. Inf. Theory, 42 (1996), 1769-1780.
doi: 10.1109/18.556673. |
[6] |
H. Chabanne, G. Cohen and A. Patey, Towards secure two-party computation from the wire-tap channel, in Information Security and Cryptology-ICISC 2013, Springer, 2013, 34-46.
doi: 10.1007/978-3-319-12160-4_3. |
[7] |
G. Cohen, S. Encheva, S. Litsyn and H.-G. Schaathun, Intersecting codes and separating codes, Discrete Appl. Math., 128 (2003), 75-83.
doi: 10.1016/S0166-218X(02)00437-7. |
[8] |
G. Cohen and A. Lempel, Linear intersecting codes, Discrete Math., 56 (1985), 35-43.
doi: 10.1016/0012-365X(85)90190-6. |
[9] |
G. Cohen, S. Mesnager and A. Patey, On minimal and quasi-minimal linear codes, in Proc. 14th Int. Conf. Crypt. Coding, Springer, Heidelberg, 2013, 85-98.
doi: 10.1007/978-3-642-45239-0_6. |
[10] |
G. Cohen and H.-G. Schaathun, Upper bounds on separating codes, IEEE Trans. Inf. Theory, 50 (2004), 1291-1295.
doi: 10.1109/TIT.2004.828140. |
[11] |
C. Ding and J. Yuan, Covering and secret sharing with linear codes, in DMTCS, Springer, 2003, 11-25.
doi: 10.1007/3-540-45066-1_2. |
[12] |
E. N. Gilbert, A comparison of signaling alphabets, Bell Syst. Techn. J., 31 (1952), 504-522. |
[13] |
F. J. MacWilliams and N. J. Sloane, The theory of error-correcting codes, North Holland, Amsterdam, 1977. |
[14] |
J. L. Massey, Minimal codewords and secret sharing, in Proc. 6th Joint Swedish-Russian Int. Workshop Info. Theory, 1993, 276-279. |
[15] |
J. L. Massey, Some applications of coding theory in cryptography, in Codes and Cyphers: Cryptography and Coding IV (ed. P.G. Farrell), 1995, 33-47. |
[16] |
H. Randriambololona, $(2,1)$-separating systems beyond the probabilistic bound, Israel J. Math., 195 (2013), 171-186.
doi: 10.1007/s11856-012-0126-9. |
[17] |
H. Randriambololona, Asymptotically good binary linear codes with asymptotically good self-intersection spans, IEEE Trans. Inf. Theory, 59 (2013), 3038-3045.
doi: 10.1109/TIT.2013.2237944. |
[18] |
H. Randriambololona, On products and powers of linear codes under componentwise multiplication, in Proc. 14th Int. Conf. Arithm. Geom. Crypt. Coding Theory (AGCT-14), Luminy, 2015, 3-7.
doi: 10.1090/conm/637/12749. |
[19] |
H. G. Schaathun, The Boneh-Shaw fingerprinting scheme is better than we thought, IEEE Trans. Inf. Forensics Sec., 1 (2006), 248-255. |
[20] |
Y. Song and Z. Li, Secret sharing with a class of minimal linear codes,, preprint, ().
|
[21] |
Y. Song, Z. Li, Y. Li and J. Li, A new multi-use multi-secret sharing scheme based on the duals of minimal linear codes, Sec. Commun. Netw., 8 (2015), 202-211. |
[22] |
M. A. Tsfasman and S. G. Vladut, Algebraic Geometric Codes, Kluwer, 1991.
doi: 10.1007/978-94-011-3810-9. |
show all references
References:
[1] |
N. Alon, G. Cohen, M. Krivilevitch and S. Litsyn, Generalized hashing and applications, JCT-A, 104 (2003), 207-215.
doi: 10.1016/j.jcta.2003.08.001. |
[2] |
A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inf. Theory, 44 (1998), 2010-2017.
doi: 10.1109/18.705584. |
[3] |
A. Ashikhmin, A. Barg, G. Cohen and L. Huguet, Variations on minimal codewords in linear codes, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer, 1995, 96-105.
doi: 10.1007/3-540-60114-7_7. |
[4] |
A. Bassa, P. Beelen, A. Garcia and H. Stichtenoth, Towers of function fields over non-prime finite fields, Moscow Math. J., 15 (2015), 1-29. |
[5] |
G. Brassard, C. Crépeau and M. Santha, Oblivious transfers and intersecting codes, IEEE Trans. Inf. Theory, 42 (1996), 1769-1780.
doi: 10.1109/18.556673. |
[6] |
H. Chabanne, G. Cohen and A. Patey, Towards secure two-party computation from the wire-tap channel, in Information Security and Cryptology-ICISC 2013, Springer, 2013, 34-46.
doi: 10.1007/978-3-319-12160-4_3. |
[7] |
G. Cohen, S. Encheva, S. Litsyn and H.-G. Schaathun, Intersecting codes and separating codes, Discrete Appl. Math., 128 (2003), 75-83.
doi: 10.1016/S0166-218X(02)00437-7. |
[8] |
G. Cohen and A. Lempel, Linear intersecting codes, Discrete Math., 56 (1985), 35-43.
doi: 10.1016/0012-365X(85)90190-6. |
[9] |
G. Cohen, S. Mesnager and A. Patey, On minimal and quasi-minimal linear codes, in Proc. 14th Int. Conf. Crypt. Coding, Springer, Heidelberg, 2013, 85-98.
doi: 10.1007/978-3-642-45239-0_6. |
[10] |
G. Cohen and H.-G. Schaathun, Upper bounds on separating codes, IEEE Trans. Inf. Theory, 50 (2004), 1291-1295.
doi: 10.1109/TIT.2004.828140. |
[11] |
C. Ding and J. Yuan, Covering and secret sharing with linear codes, in DMTCS, Springer, 2003, 11-25.
doi: 10.1007/3-540-45066-1_2. |
[12] |
E. N. Gilbert, A comparison of signaling alphabets, Bell Syst. Techn. J., 31 (1952), 504-522. |
[13] |
F. J. MacWilliams and N. J. Sloane, The theory of error-correcting codes, North Holland, Amsterdam, 1977. |
[14] |
J. L. Massey, Minimal codewords and secret sharing, in Proc. 6th Joint Swedish-Russian Int. Workshop Info. Theory, 1993, 276-279. |
[15] |
J. L. Massey, Some applications of coding theory in cryptography, in Codes and Cyphers: Cryptography and Coding IV (ed. P.G. Farrell), 1995, 33-47. |
[16] |
H. Randriambololona, $(2,1)$-separating systems beyond the probabilistic bound, Israel J. Math., 195 (2013), 171-186.
doi: 10.1007/s11856-012-0126-9. |
[17] |
H. Randriambololona, Asymptotically good binary linear codes with asymptotically good self-intersection spans, IEEE Trans. Inf. Theory, 59 (2013), 3038-3045.
doi: 10.1109/TIT.2013.2237944. |
[18] |
H. Randriambololona, On products and powers of linear codes under componentwise multiplication, in Proc. 14th Int. Conf. Arithm. Geom. Crypt. Coding Theory (AGCT-14), Luminy, 2015, 3-7.
doi: 10.1090/conm/637/12749. |
[19] |
H. G. Schaathun, The Boneh-Shaw fingerprinting scheme is better than we thought, IEEE Trans. Inf. Forensics Sec., 1 (2006), 248-255. |
[20] |
Y. Song and Z. Li, Secret sharing with a class of minimal linear codes,, preprint, ().
|
[21] |
Y. Song, Z. Li, Y. Li and J. Li, A new multi-use multi-secret sharing scheme based on the duals of minimal linear codes, Sec. Commun. Netw., 8 (2015), 202-211. |
[22] |
M. A. Tsfasman and S. G. Vladut, Algebraic Geometric Codes, Kluwer, 1991.
doi: 10.1007/978-94-011-3810-9. |
[1] |
Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Addendum. Advances in Mathematics of Communications, 2011, 5 (3) : 543-546. doi: 10.3934/amc.2011.5.543 |
[2] |
Irene Márquez-Corbella, Edgar Martínez-Moro. Algebraic structure of the minimal support codewords set of some linear codes. Advances in Mathematics of Communications, 2011, 5 (2) : 233-244. doi: 10.3934/amc.2011.5.233 |
[3] |
Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 2021, 15 (2) : 227-240. doi: 10.3934/amc.2020055 |
[4] |
Gianira N. Alfarano, Martino Borello, Alessandro Neri. A geometric characterization of minimal codes and their asymptotic performance. Advances in Mathematics of Communications, 2022, 16 (1) : 115-133. doi: 10.3934/amc.2020104 |
[5] |
Nuh Aydin, Nicholas Connolly, Markus Grassl. Some results on the structure of constacyclic codes and new linear codes over GF(7) from quasi-twisted codes. Advances in Mathematics of Communications, 2017, 11 (1) : 245-258. doi: 10.3934/amc.2017016 |
[6] |
Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Advances in Mathematics of Communications, 2010, 4 (1) : 69-81. doi: 10.3934/amc.2010.4.69 |
[7] |
Petr Lisoněk, Layla Trummer. Algorithms for the minimum weight of linear codes. Advances in Mathematics of Communications, 2016, 10 (1) : 195-207. doi: 10.3934/amc.2016.10.195 |
[8] |
Jean Creignou, Hervé Diet. Linear programming bounds for unitary codes. Advances in Mathematics of Communications, 2010, 4 (3) : 323-344. doi: 10.3934/amc.2010.4.323 |
[9] |
Fernando Hernando, Tom Høholdt, Diego Ruano. List decoding of matrix-product codes from nested codes: An application to quasi-cyclic codes. Advances in Mathematics of Communications, 2012, 6 (3) : 259-272. doi: 10.3934/amc.2012.6.259 |
[10] |
Nigel Boston, Jing Hao. The weight distribution of quasi-quadratic residue codes. Advances in Mathematics of Communications, 2018, 12 (2) : 363-385. doi: 10.3934/amc.2018023 |
[11] |
Enhui Lim, Frédérique Oggier. On the generalised rank weights of quasi-cyclic codes. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022010 |
[12] |
Fernando Hernando, Diego Ruano. New linear codes from matrix-product codes with polynomial units. Advances in Mathematics of Communications, 2010, 4 (3) : 363-367. doi: 10.3934/amc.2010.4.363 |
[13] |
Ettore Fornasini, Telma Pinho, Raquel Pinto, Paula Rocha. Composition codes. Advances in Mathematics of Communications, 2016, 10 (1) : 163-177. doi: 10.3934/amc.2016.10.163 |
[14] |
John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019 |
[15] |
Peter Vandendriessche. LDPC codes associated with linear representations of geometries. Advances in Mathematics of Communications, 2010, 4 (3) : 405-417. doi: 10.3934/amc.2010.4.405 |
[16] |
Ali Tebbi, Terence Chan, Chi Wan Sung. Linear programming bounds for distributed storage codes. Advances in Mathematics of Communications, 2020, 14 (2) : 333-357. doi: 10.3934/amc.2020024 |
[17] |
Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044 |
[18] |
Thomas Feulner. Canonization of linear codes over $\mathbb Z$4. Advances in Mathematics of Communications, 2011, 5 (2) : 245-266. doi: 10.3934/amc.2011.5.245 |
[19] |
Tatsuya Maruta, Yusuke Oya. On optimal ternary linear codes of dimension 6. Advances in Mathematics of Communications, 2011, 5 (3) : 505-520. doi: 10.3934/amc.2011.5.505 |
[20] |
Vito Napolitano, Ferdinando Zullo. Codes with few weights arising from linear sets. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020129 |
2020 Impact Factor: 0.935
Tools
Metrics
Other articles
by authors
[Back to Top]