Hermitian Self-Dual, MDS, and Generalized Reed–Solomon Codes

Both self-dual codes and maximum distance separable (MDS) codes have nice algebraic structures, theoretical significance, and practical implications. We present two classes of the Hermitian self-dual, MDS, and generalized Reed–Solomon (GRS) codes. Conversely, we prove that the Hermitian self-dual, MDS, and GRS codes must be above two classes for <inline-formula> <tex-math notation="LaTeX">${n =4}$ </tex-math></inline-formula> and conjecture that this result also holds for each even length <inline-formula> <tex-math notation="LaTeX">$ {n}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$4\le {n}\le \text {q+1}$ </tex-math></inline-formula>.

[1]  Qin Yue,et al.  Some quantum MDS codes with large minimum distance from generalized Reed-Solomon codes , 2017, Cryptography and Communications.

[2]  Patric R. J. Östergård,et al.  On the Classification of MDS Codes , 2014, IEEE Transactions on Information Theory.

[3]  Xianmang He,et al.  New q-ary Quantum MDS Codes with Distances Bigger than q/2 , 2015, ArXiv.

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

[5]  Chau Yuen,et al.  On the existence of MDS codes over small fields with constrained generator matrices , 2014, 2014 IEEE International Symposium on Information Theory.

[6]  Jon-Lark Kim,et al.  MDS self-dual codes , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[7]  Jinquan Luo,et al.  Construction of MDS Self-dual Codes over Finite Fields , 2019, Finite Fields Their Appl..

[8]  Ron M. Roth,et al.  A construction of non-Reed-Solomon type MDS codes , 1989, IEEE Trans. Inf. Theory.

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

[10]  Carsten Dahl,et al.  Classification of pseudo-cyclic MDS codes , 1991, IEEE Trans. Inf. Theory.

[11]  Qin Yue,et al.  Construction of some new quantum MDS codes , 2017, Finite Fields Their Appl..

[12]  Jon-Lark Kim,et al.  New MDS or Near-MDS Self-Dual Codes , 2008, IEEE Transactions on Information Theory.

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

[14]  Jon-Lark Kim,et al.  Euclidean and Hermitian self-dual MDS codes over large finite fields , 2004, J. Comb. Theory, Ser. A.

[15]  Chaoping Xing,et al.  A Construction of New Quantum MDS Codes , 2013, IEEE Transactions on Information Theory.

[16]  Haibin Kan,et al.  Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes , 2017, Des. Codes Cryptogr..

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

[18]  Jinquan Luo,et al.  New MDS self-dual codes over finite fields of odd characteristic , 2020, Des. Codes Cryptogr..

[19]  Xianmang He,et al.  New q-ary quantum MDS codes with distances bigger than $$\frac{q}{2}$$q2 , 2016, Quantum Inf. Process..

[20]  S. Georgiou,et al.  MDS Self-Dual Codes over Large Prime Fields , 2002 .