A hierarchical poset code’s hull is defined as its intersection with its dual. Linear codes with small dimensions have significant potential for applications in algorithmic aspects of codes in classical as well as quantum coding theory. This article examines the hull of a binary hierarchical poset code whose dimension is expressed in terms of the rank of the Gramians of its generator matrix. Additionally, an upper bound on (A_q(n, k, h_dim)) - the maximum distance d of an (n, k) hierarchical poset-code with hull dimension h_dim - is derived using Griesmer’s bound. Furthermore, we investigate the construction and optimality of two dimensional binary hierarchical poset code having hull dimension one.

hierarchical poset code, hull, Griesmer bound, optimal code

Received: January 4, 2024; Accepted: February 21, 2024; Published: March 21, 2024

How to cite this article: Rohini Baliram More and Venkatrajam Marka, Optimal binary hierarchical poset code having hull dimension one, Advances and Applications in Discrete Mathematics 41(3) (2024), 239-259. http://dx.doi.org/10.17654/0974165824018

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] E. F. Assmus and J. D. Key, Affine and projective plane, Discrete Mathematics 83 (1990), 161-187.
[2] N. Sendrier, Finding the permutation between equivalent linear codes: the support splitting algorithm, IEEE Transactions on Information Theory 46(4) (2000), 1193-1203. https://doi.org/10.1109/18.850662.
[3] J. S. Leon, Computing automorphism groups of error-correcting codes, IEEE Transactions on Information Theory 28(3) (1982), 496-511.
https://doi.org/10. 1109/TIT.1982.1056498.
[4] N. Sendrier and G. Skersys, On the computation of the automorphism group of a linear code, IEEE International Symposium on Information Theory - Proceedings, 13 (2001). https://doi.org/10.1109/ISIT.2001.935876.
[5] K. Guenda, S. Jitman and T. A. Gulliver, Constructions of good entanglement-assisted quantum error correcting codes, Designs, Codes, and Cryptography 86(1) (2018), 121-136. https://doi.org/10.1007/s10623-017-0330-z.
[6] G. Luo, X. Cao and X. Chen, MDS codes with hulls of arbitrary dimensions and their quantum error correction, IEEE Transactions on Information Theory 65(5) (2019), 2944-2952. https://doi.org/10.1109/TIT.2018.2874953.
[7] S. Thipworawimon and S. Jitman, Hulls of linear codes revisited with applications, Journal of Applied Mathematics and Computing 62(1-2) (2020), 325-340. https://doi.org/10.1007/s12190-019-01286-7.
[8] L. Sok, MDS linear codes with one-dimensional hull, Cryptography and Communications 14(5) (2022), 949-971.
https://doi.org/10.1007/s12095-022-00559-6.
[9] W. Fang, F. W. Fu, L. Li and S. Zhu, Euclidean and Hermitian Hulls of MDS codes and their applications to EAQECCs, IEEE Transactions on Information Theory 66(6) (2020), 3527-3537. https://doi.org/10.1109/TIT.2019.2950245.
[10] N. Gao, J. Li and S. Huang, Hermitian Hulls of constacyclic codes and their applications to quantum codes, International Journal of Theoretical Physics 61(57) (2022), 1-14. https://doi.org/10.1007/s10773-022-05012-1.
[11] N. Sendrier, On the dimension of the hull, SIAM Journal on Discrete Mathematics 10(2) (1997), 282-293. https://doi.org/10.1137/S0895480195294027.
[12] C. Li and P. Zeng, Constructions of linear codes with one-dimensional hull, IEEE Transactions on Information Theory 65(3) (2019), 1668-1676.
https://doi.org/10. 1109/TIT.2018.2863693.
[13] C. Carlet, C. Li and S. Mesnager, Linear codes with small hulls in semi-primitive case, Designs, Codes, and Cryptography 87(12) (2019), 3063-3075.
https://doi.org/10.1007/s10623-019-00663-4.
[14] L. Qian, X. Cao and S. Mesnager, Linear codes with one-dimensional hull associated with Gaussian sums, Cryptography and Communications 13(2) (2021), 225-243. https://doi.org/10.1007/s12095-020-00462-y.
[15] Y. Wang and R. Tao, Constructions of linear codes with small hulls from association schemes, Advances in Mathematics of Communications 16(2) (2022), 349-364. https://doi.org/10.3934/amc.2020114.
[16] L. Sok, On linear codes with one-dimensional Euclidean hull and their applications to EAQECCs, IEEE Transactions on Information Theory 68(7) (2022), 4329-4343. https://doi.org/10.1109/TIT.2022.3152580.
arXiv:2101.06461.
[17] H. Liu and X. Pan, Galois hulls of linear codes over finite fields, Designs, Codes, and Cryptography 88(2) (2020), 241-255.
https://doi.org/10.1007/s10623-019-00681-2. arXiv:1809.08053.
[18] X. Fang, R. Jin, J. Luo and W. Ma, New Galois Hulls of GRS codes and application to EAQECCs, Cryptography and Communications 14(1) (2022), 145-159. https://doi.org/10.1007/s12095-021-00525-8.
[19] T. Mankean and S. Jitman, Optimal binary and ternary linear codes with hull dimension one, Journal of Applied Mathematics and Computing 64(1-2) (2020), 137-155. https://doi.org/10.1007/s12190-020-01348-1.
[20] T. Mankean and S. Jitman, Constructions and bounds on quaternary linear codes with Hermitian hull dimension one, Arabian Journal of Mathematics 10(1) (2021), 175-184. https://doi.org/10.1007/s40065-020-00303-z.
[21] R. A. Brualdi, J. S. Graves and K. M. Lawrence, Codes with a poset metric, Discrete Mathematics 147(1-3) (1995), 57-72.
https://doi.org/10.1016/0012-365X(94)00228-B.
[22] M. Firer, M. Muniz, B. Steinberg and G. Yin, Poset codes : partial orders, Metrics and Coding Theory 1 (2018), 1-127. https://doi.org/10.1007/978-3-319-93821-9.
[23] R. A. Machado, J. A. Pinheiro and M. Firer, Characterization of metrics induced by hierarchical posets, IEEE Transactions on Information Theory 63(6) (2017), 3630-3640. https://doi.org/10.1109/TIT.2017.2691763. arXiv:1508.00914.
[24] L. V. Felix and M. Firer, Canonical-systematic form for codes in hierarchical poset metrics, Advances in Mathematics of Communications 6(3) (2012), 315-328. https://doi.org/10.3934/amc.2012.6.315.
[25] W. C. Huffman and V. Pless, Fundamental of Error-correcting Codes, Cambridge, University Press, 2003.