On Linear Codes With One-Dimensional Euclidean Hull and Their Applications to EAQECCs

The Euclidean hull of a linear code C is the intersection of C with its Euclidean dual C. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes. Several classes of optimal linear codes with one-dimensional Euclidean hull are constructed. Some new EAQECCs are presented.

[1]  Fang-Wei Fu,et al.  New Constructions of MDS Euclidean Self-Dual Codes From GRS Codes and Extended GRS Codes , 2019, IEEE Transactions on Information Theory.

[2]  Gintaras Skersys,et al.  The Average Dimension of the Hull of Cyclic Codes , 2001, Discret. Appl. Math..

[3]  Garry Bowen.  Entanglement required in achieving entanglement-assisted channel capacities , 2002 .

[4]  Henning Stichtenoth,et al.  Algebraic function fields and codes , 1993, Universitext.

[5]  Alexei E. Ashikhmin,et al.  Nonbinary quantum stabilizer codes , 2001, IEEE Trans. Inf. Theory.

[6]  Haode Yan.  A note on the constructions of MDS self-dual codes , 2018, Cryptography and Communications.

[7]  Igor Devetak,et al.  Correcting Quantum Errors with Entanglement , 2006, Science.

[8]  Jennifer D. Key,et al.  Affine and projective planes , 1990, Discret. Math..

[9]  Lingfei Jin.  Construction of MDS Codes With Complementary Duals , 2017, IEEE Transactions on Information Theory.

[10]  Xiaojing Chen,et al.  MDS Codes With Hulls of Arbitrary Dimensions and Their Quantum Error Correction , 2019, IEEE Transactions on Information Theory.

[11]  Sihem Mesnager,et al.  On σ-LCD codes , 2017, ArXiv.

[12]  Alexei Ashikhmin,et al.  Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators , 2016, IEEE Transactions on Information Theory.

[13]  Haode Yan,et al.  Parameters of LCD BCH codes with two lengths , 2018, Adv. Math. Commun..

[14]  Lingfei Jin,et al.  Explicit MDS Codes With Complementary Duals , 2017, IEEE Transactions on Information Theory.

[15]  Sihem Mesnager,et al.  Euclidean and Hermitian LCD MDS codes , 2017, Des. Codes Cryptogr..

[16]  V. D. Goppa.  ALGEBRAICO-GEOMETRIC CODES , 1983 .

[17]  Cunsheng Ding,et al.  LCD Cyclic Codes Over Finite Fields , 2017, IEEE Transactions on Information Theory.

[18]  Cunsheng Ding,et al.  Two Families of LCD BCH Codes , 2016, IEEE Transactions on Information Theory.

[19]  Sihem Mesnager,et al.  Linear Codes Over 𝔽q Are Equivalent to LCD Codes for q>3 , 2018, IEEE Trans. Inf. Theory.

[20]  Xiang Yang,et al.  The condition for a cyclic code to have a complementary dual , 1994, Discret. Math..

[21]  Carlos Galindo,et al.  Entanglement-assisted quantum error-correcting codes over arbitrary finite fields , 2018, Quantum Information Processing.

[22]  Sihem Mesnager,et al.  Linear codes with small hulls in semi-primitive case , 2019, Designs, Codes and Cryptography.

[23]  Chaoping Xing,et al.  New MDS Self-Dual Codes From Generalized Reed—Solomon Codes , 2016, IEEE Transactions on Information Theory.

[24]  James L. Massey,et al.  Linear codes with complementary duals , 1992, Discret. Math..

[25]  JM Jeroen Doumen,et al.  Some applications of coding theory in cryptography , 2003 .

[26]  Claude Carlet,et al.  On Linear Complementary Pairs of Codes , 2018, IEEE Transactions on Information Theory.

[27]  T. Brun,et al.  Entanglement increases the error-correcting ability of quantum error-correcting codes , 2010, 1008.2598.

[28]  Jun Heo,et al.  Entanglement-assisted codeword stabilized quantum codes , 2011 .

[29]  Peng Zeng,et al.  Constructions of Linear Codes With One-Dimensional Hull , 2019, IEEE Transactions on Information Theory.

[30]  Sihem Mesnager,et al.  Complementary Dual Algebraic Geometry Codes , 2016, IEEE Transactions on Information Theory.

[31]  Markus Grassl,et al.  On self-dual MDS codes , 2008, 2008 IEEE International Symposium on Information Theory.

[32]  Sihem Mesnager,et al.  New Characterization and Parametrization of LCD Codes , 2019, IEEE Transactions on Information Theory.

[33]  Mario A. de Boer,et al.  Almost MDS codes , 1996, Des. Codes Cryptogr..

[34]  Claude Carlet,et al.  Complementary dual codes for counter-measures to side-channel attacks , 2016, Adv. Math. Commun..

[35]  Jeffrey S. Leon,et al.  Computing automorphism groups of error-correcting codes , 1982, IEEE Trans. Inf. Theory.

[36]  Bocong Chen,et al.  New Constructions of MDS Codes With Complementary Duals , 2017, IEEE Transactions on Information Theory.

[37]  Shudi Yang,et al.  New LCD MDS codes constructed from generalized Reed–Solomon codes , 2019, Journal of Algebra and Its Applications.

[38]  Lin Sok,et al.  MDS linear codes with one-dimensional hull , 2020, Cryptography and Communications.

[39]  J. S. Leon,et al.  Permutation Group Algorithms Based on Partitions, I: Theory and Algorithms , 1991, J. Symb. Comput..

[40]  Sihem Mesnager,et al.  Linear codes with one-dimensional hull associated with Gaussian sums , 2020, Cryptography and Communications.

[41]  Nicolas Sendrier,et al.  On the Dimension of the Hull , 1997, SIAM J. Discret. Math..

[42]  W. Bosma,et al.  HANDBOOK OF MAGMA FUNCTIONS , 2011 .

[43]  Xiaoqing Wang,et al.  New MDS Euclidean and Hermitian self-dual codes over finite fields , 2016, ArXiv.

[44]  Ivan N. Landjev,et al.  Near-MDS codes over some small fields , 2000, Discret. Math..

[45]  T. Brun,et al.  Optimal entanglement formulas for entanglement-assisted quantum coding , 2008, 0804.1404.

[46]  Patanee Udomkavanich,et al.  Hulls of cyclic and negacyclic codes over finite fields , 2015, Finite Fields Their Appl..

[47]  Lin Sok.  Explicit Constructions of MDS Self-Dual Codes , 2020, IEEE Transactions on Information Theory.

[48]  Kenza Guenda.  New MDS self-dual codes over finite fields , 2012, Des. Codes Cryptogr..