# American Institute of Mathematical Sciences

February  2016, 10(1): 53-61. doi: 10.3934/amc.2016.10.53

## 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

Received  December 2014 Revised  May 2015 Published  March 2016

Minimal linear codes are linear codes such that the support of every codeword does not contain the support of another linearly independent codeword. Such codes have applications in cryptography, e.g. to secret sharing. We pursue here their study and construct improved asymptotically good families of minimal linear codes. We also consider quasi-minimal, $t$-minimal, and $t$-quasi-minimal linear codes, which are new variations on this notion.
Citation: Gérard Cohen, Sihem Mesnager, Hugues Randriam. Yet another variation on minimal linear codes. Advances in Mathematics of Communications, 2016, 10 (1) : 53-61. doi: 10.3934/amc.2016.10.53
##### References:
 [1] N. Alon, G. Cohen, M. Krivilevitch and S. Litsyn, Generalized hashing and applications,, JCT-A, 104 (2003), 207. doi: 10.1016/j.jcta.2003.08.001. Google Scholar [2] A. Ashikhmin and A. Barg, Minimal vectors in linear codes,, IEEE Trans. Inf. Theory, 44 (1998), 2010. doi: 10.1109/18.705584. Google Scholar [3] A. Ashikhmin, A. Barg, G. Cohen and L. Huguet, Variations on minimal codewords in linear codes,, in Applied Algebra, (1995), 96. doi: 10.1007/3-540-60114-7_7. Google Scholar [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. Google Scholar [5] G. Brassard, C. Crépeau and M. Santha, Oblivious transfers and intersecting codes,, IEEE Trans. Inf. Theory, 42 (1996), 1769. doi: 10.1109/18.556673. Google Scholar [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, (2013), 34. doi: 10.1007/978-3-319-12160-4_3. Google Scholar [7] G. Cohen, S. Encheva, S. Litsyn and H.-G. Schaathun, Intersecting codes and separating codes,, Discrete Appl. Math., 128 (2003), 75. doi: 10.1016/S0166-218X(02)00437-7. Google Scholar [8] G. Cohen and A. Lempel, Linear intersecting codes,, Discrete Math., 56 (1985), 35. doi: 10.1016/0012-365X(85)90190-6. Google Scholar [9] G. Cohen, S. Mesnager and A. Patey, On minimal and quasi-minimal linear codes,, in Proc. 14th Int. Conf. Crypt. Coding, (2013), 85. doi: 10.1007/978-3-642-45239-0_6. Google Scholar [10] G. Cohen and H.-G. Schaathun, Upper bounds on separating codes,, IEEE Trans. Inf. Theory, 50 (2004), 1291. doi: 10.1109/TIT.2004.828140. Google Scholar [11] C. Ding and J. Yuan, Covering and secret sharing with linear codes,, in DMTCS, (2003), 11. doi: 10.1007/3-540-45066-1_2. Google Scholar [12] E. N. Gilbert, A comparison of signaling alphabets,, Bell Syst. Techn. J., 31 (1952), 504. Google Scholar [13] F. J. MacWilliams and N. J. Sloane, The theory of error-correcting codes,, North Holland, (1977). Google Scholar [14] J. L. Massey, Minimal codewords and secret sharing,, in Proc. 6th Joint Swedish-Russian Int. Workshop Info. Theory, (1993), 276. Google Scholar [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. Google Scholar [16] H. Randriambololona, $(2,1)$-separating systems beyond the probabilistic bound,, Israel J. Math., 195 (2013), 171. doi: 10.1007/s11856-012-0126-9. Google Scholar [17] H. Randriambololona, Asymptotically good binary linear codes with asymptotically good self-intersection spans,, IEEE Trans. Inf. Theory, 59 (2013), 3038. doi: 10.1109/TIT.2013.2237944. Google Scholar [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), (2015), 3. doi: 10.1090/conm/637/12749. Google Scholar [19] H. G. Schaathun, The Boneh-Shaw fingerprinting scheme is better than we thought,, IEEE Trans. Inf. Forensics Sec., 1 (2006), 248. Google Scholar [20] Y. Song and Z. Li, Secret sharing with a class of minimal linear codes,, preprint, (). Google Scholar [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. Google Scholar [22] M. A. Tsfasman and S. G. Vladut, Algebraic Geometric Codes,, Kluwer, (1991). doi: 10.1007/978-94-011-3810-9. Google Scholar

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##### References:
 [1] N. Alon, G. Cohen, M. Krivilevitch and S. Litsyn, Generalized hashing and applications,, JCT-A, 104 (2003), 207. doi: 10.1016/j.jcta.2003.08.001. Google Scholar [2] A. Ashikhmin and A. Barg, Minimal vectors in linear codes,, IEEE Trans. Inf. Theory, 44 (1998), 2010. doi: 10.1109/18.705584. Google Scholar [3] A. Ashikhmin, A. Barg, G. Cohen and L. Huguet, Variations on minimal codewords in linear codes,, in Applied Algebra, (1995), 96. doi: 10.1007/3-540-60114-7_7. Google Scholar [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. Google Scholar [5] G. Brassard, C. Crépeau and M. Santha, Oblivious transfers and intersecting codes,, IEEE Trans. Inf. Theory, 42 (1996), 1769. doi: 10.1109/18.556673. Google Scholar [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, (2013), 34. doi: 10.1007/978-3-319-12160-4_3. Google Scholar [7] G. Cohen, S. Encheva, S. Litsyn and H.-G. Schaathun, Intersecting codes and separating codes,, Discrete Appl. Math., 128 (2003), 75. doi: 10.1016/S0166-218X(02)00437-7. Google Scholar [8] G. Cohen and A. Lempel, Linear intersecting codes,, Discrete Math., 56 (1985), 35. doi: 10.1016/0012-365X(85)90190-6. Google Scholar [9] G. Cohen, S. Mesnager and A. Patey, On minimal and quasi-minimal linear codes,, in Proc. 14th Int. Conf. Crypt. Coding, (2013), 85. doi: 10.1007/978-3-642-45239-0_6. Google Scholar [10] G. Cohen and H.-G. Schaathun, Upper bounds on separating codes,, IEEE Trans. Inf. Theory, 50 (2004), 1291. doi: 10.1109/TIT.2004.828140. Google Scholar [11] C. Ding and J. Yuan, Covering and secret sharing with linear codes,, in DMTCS, (2003), 11. doi: 10.1007/3-540-45066-1_2. Google Scholar [12] E. N. Gilbert, A comparison of signaling alphabets,, Bell Syst. Techn. J., 31 (1952), 504. Google Scholar [13] F. J. MacWilliams and N. J. Sloane, The theory of error-correcting codes,, North Holland, (1977). Google Scholar [14] J. L. Massey, Minimal codewords and secret sharing,, in Proc. 6th Joint Swedish-Russian Int. Workshop Info. Theory, (1993), 276. Google Scholar [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. Google Scholar [16] H. Randriambololona, $(2,1)$-separating systems beyond the probabilistic bound,, Israel J. Math., 195 (2013), 171. doi: 10.1007/s11856-012-0126-9. Google Scholar [17] H. Randriambololona, Asymptotically good binary linear codes with asymptotically good self-intersection spans,, IEEE Trans. Inf. Theory, 59 (2013), 3038. doi: 10.1109/TIT.2013.2237944. Google Scholar [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), (2015), 3. doi: 10.1090/conm/637/12749. Google Scholar [19] H. G. Schaathun, The Boneh-Shaw fingerprinting scheme is better than we thought,, IEEE Trans. Inf. Forensics Sec., 1 (2006), 248. Google Scholar [20] Y. Song and Z. Li, Secret sharing with a class of minimal linear codes,, preprint, (). Google Scholar [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. Google Scholar [22] M. A. Tsfasman and S. G. Vladut, Algebraic Geometric Codes,, Kluwer, (1991). doi: 10.1007/978-94-011-3810-9. Google Scholar
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