On codes induced from Hadamard matrices

Ted Hurley; Ted Hurley

doi:10.3934/math.20251243

AIMS Mathematics

2025, Volume 10, Issue 12: 28264-28276. doi: 10.3934/math.20251243

Previous Article Next Article

Research article Special Issues

On codes induced from Hadamard matrices

Ted Hurley ^,

University of Galway (previously National University of Ireland Galway), Ireland

Received: 09 September 2025 Revised: 21 October 2025 Accepted: 10 November 2025 Published: 01 December 2025
MSC : 15B99, 94B05, 94B10

Unit derived schemes applied to Hadamard matrices are used to construct and analyse linear block and convolutional codes. Codes are constructed to prescribed types, lengths and rates and multiple series of self-dual, dual-containing, linear complementary dual and quantum error-correcting of both linear block and convolutional codes are derived.
- Hadamard,
- code,
- linear,
- convolutional,
- self-dual,
- dual-containing,
- quantum code
Citation: Ted Hurley. On codes induced from Hadamard matrices[J]. AIMS Mathematics, 2025, 10(12): 28264-28276. doi: 10.3934/math.20251243

Related Papers:

Abstract

Unit derived schemes applied to Hadamard matrices are used to construct and analyse linear block and convolutional codes. Codes are constructed to prescribed types, lengths and rates and multiple series of self-dual, dual-containing, linear complementary dual and quantum error-correcting of both linear block and convolutional codes are derived.

References

[1]	P. Almeida, D. Napp, R. Pinto, A new class of superregular matrices and MDP convolutional codes, Linear Algebra Appl., 439 (2013), 2145–2157. https://doi.org/10.1016/j.laa.2013.06.013 doi: 10.1016/j.laa.2013.06.013
[2]	P. J. Almeida, D. Napp, R. Pinto, Superregular matrices and applications to convolutional codes, Linear Algebra Appl., 499 (2016), 1–25. https://doi.org/10.1016/j.laa.2016.02.034 doi: 10.1016/j.laa.2016.02.034
[3]	R. E. Blahut, Algebraic Codes for data transmission, Cambridge University Press, 2012. https://doi.org/10.1017/CBO9780511800467
[4]	I. Bocharova, F. Hug, R. Johannesson, B. D. Kudryashov, Dual convolutional codes and the MacWilliams identities, Probl. Inf. Transm., 48 (2012), 21–30. https://doi.org/10.1134/S0032946012010036 doi: 10.1134/S0032946012010036
[5]	A. R. Calderbank, E. M. Rains, P. M. Shor, N. J. A. Sloane, Quantum error correction via codes over $GF(4)$, IEEE Trans. Inform. Theory, 44 (1998), 1369–1387. https://doi.org/10.1109/18.681315 doi: 10.1109/18.681315
[6]	A. R. Calderbank, P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A, 54 (1996), 1098–1105. https://doi.org/10.1103/PhysRevA.54.1098 doi: 10.1103/PhysRevA.54.1098
[7]	C. Carlet, S. Mesnager, C. Tang, Y. Qi, Euclidean and Hermitian LCD MDS codes, Des. Codes Cryptogr., 86 (2018), 2605–2618. https://doi.org/10.1007/s10623-018-0463-8 doi: 10.1007/s10623-018-0463-8
[8]	C. Carlet, S. Mesnager, C. Tang, Y. Qi, R. Pelikaan, Linear codes over $F_q$ are equivalent to LCD codes for $q>3$, IEEE Trans. Inform. Theory, 64 (2018), 3010–3017. https://doi.org/10.1109/TIT.2018.2789347 doi: 10.1109/TIT.2018.2789347
[9]	C. Carlet, S. Mesnager, C. Tang, Y. Qi, New characterization and parametrization of LCD codes, IEEE Trans. Inform. Theory, 65 (2019), 39–49. https://doi.org/10.1109/TIT.2018.2829873 doi: 10.1109/TIT.2018.2829873
[10]	C. Carlet, Boolean functions for cryptography and error correcting codes, In: Boolean models and methods in mathematics, computer science, and engineering, Cambridge University Pres, 2010,257–397. https://doi.org/10.1017/CBO9780511780448.011
[11]	C. Carlet, S. Guilley, Complementary dual codes for counter measures to side-channel attacks, In: Coding theory and applications, Cham: Springer, 3 (2015), 97–105. https://doi.org/10.1007/978-3-319-17296-5_9
[12]	H. Gluesing-Luerssen, G. Schneider, A MacWilliams identity for convolutional codes: The general case, IEEE Trans. Inform. Theory, 55 (2009), 2920–2930. https://doi.org/10.1109/TIT.2009.2021302 doi: 10.1109/TIT.2009.2021302
[13]	G. G. La Guardia, On negacyclic MDS-convolutiona codes, Linear Algebra Appl., 448 (2014), 85–96. https://doi.org/10.1016/j.laa.2014.01.033 doi: 10.1016/j.laa.2014.01.033
[14]	P. Hurley, T. Hurley, Module codes in group rings, In: 2007 IEEE international symposium on information theory, Nice: IEEE, 2007, 1981–1985. https://doi.org/10.1109/ISIT.2007.4557511
[15]	P. Hurley, T. Hurley, Codes from zero-divisors and units in group rings, Int. J. Inf. Coding Theory, 1 (2009), 57–87. https://doi.org/10.1504/IJICoT.2009.024047 doi: 10.1504/IJICoT.2009.024047
[16]	P. Hurley, T. Hurley, Block codes from matrix and group rings, In: Selected topics in information and coding theory, World Scientific Publishing Co., 2010,159–194. https://doi.org/10.1142/9789812837172_0005
[17]	P. Hurley, T. Hurley, LDPC and convolutional codes from matrix and group rings, In: Selected topics in information and coding theory, World Scientific Publishing Co., 2010,195–237. https://doi.org/10.1142/9789812837172_0006
[18]	K. J. Horadam, Hadamard matrices and their applications, Princeton University Press, 2007.
[19]	T. Hurley, D. Hurley, Coding theory: The unit-derived methodology, Int. J. Inf. Coding Theory, 5 (2018), 55–80. https://doi.org/10.1504/IJICOT.2018.10013082 doi: 10.1504/IJICOT.2018.10013082
[20]	T. Hurley, D. Hurley, B. Hurley, Quantum error-correcting codes: The unit design strategy, Int. J. Inf. Coding Theory, 5 (2018), 169–182. https://doi.org/10.1504/IJICOT.2018.095018 doi: 10.1504/IJICOT.2018.095018
[21]	T. Hurley, D. Hurley, B. Hurley, Maximum distance separable codes to order, arXiv: 1902.06624, 2019. https://doi.org/10.48550/arXiv.1902.06624
[22]	T. Hurley, Ultimate linear block and convolutional codes, AIMS Mathematics, 10 (2025), 8398–8421. https://doi.org/10.3934/math.2025387 doi: 10.3934/math.2025387
[23]	T. Hurley, Linear block and convolutional MDS codes to equired rate, distance and type, In: Intelligent computing, Cham: Springer, 507 (2022), 129–157. https://doi.org/10.1007/978-3-031-10464-0_10
[24]	R. Johannesson, K. Sh. Zigangirov, Fundamentals of convolutional coding, Wiley-IEEE Press, 2015.
[25]	J. H. van Lint, R. M. Wilson, A course in combinatorics, Cambridge University Press, 2001.
[26]	F. J. MacWilliams, N. J. A. Sloane, The theory of error-correcting codes, Elsevier, 1977.
[27]	C. L. Mallows, N. J. A. Sloan, An upper bound for self-dual codes, Inf. Control, 22 (1973), 188–200. https://doi.org/10.1016/S0019-9958(73)90273-8 doi: 10.1016/S0019-9958(73)90273-8
[28]	J. L. Massey, Reversible codes, Inf. Control, 7 (1964), 369–380. https://doi.org/10.1016/S0019-9958(64)90438-3 doi: 10.1016/S0019-9958(64)90438-3
[29]	J. L. Massey, Linear codes with complementary duals, Discrete Math., 106-107 (1992), 337–342. https://doi.org/10.1016/0012-365X(92)90563-U doi: 10.1016/0012-365X(92)90563-U
[30]	R. J. McEliece, The theory of information and coding, 2 Eds., Cambridge University Press, 2002. https://doi.org/10.1017/CBO9780511606267
[31]	S. Mesnager, C. Tang, Y. Qi, Complementary dual algebraic geometry codes, IEEE Trans. Inform. Theory, 64 (2018), 2390–2397. https://doi.org/10.1109/TIT.2017.2766075 doi: 10.1109/TIT.2017.2766075
[32]	V. Pless, Symmetry codes over GF(3) and new five-designs, J. Combin. Theory Ser. A, 12 (1972), 119–142. https://doi.org/10.1016/0097-3165(72)90088-X doi: 10.1016/0097-3165(72)90088-X
[33]	J. M. M. Porras, J. A. D. Pérez, J. I. I. Curto, G. S. Sotelo, Convolutional goppa codes, IEEE Trans. Inf. Theory, 52 (2006), 340–344. https://doi.org/10.1109/TIT.2005.860447 doi: 10.1109/TIT.2005.860447
[34]	J. Rosenthal, R. Smarandache, Maximum distance separable convolutional codes, Appl. Algebra Engrg. Comm. Comput., 10 (1999), 15–32. https://doi.org/10.1007/s002000050120 doi: 10.1007/s002000050120
[35]	J. Rosenthal, Connections between linear systems and convolutional codes, In: Codes, systems, and graphical models, New York: Springer, 123 (2001), 39–66. https://doi.org/10.1007/978-1-4613-0165-3_2
[36]	A. M. Steane, Simple quantum error-correcting codes, Phys. Rev. A, 54 (1996), 4741–4751. https://doi.org/10.1103/PhysRevA.54.4741 doi: 10.1103/PhysRevA.54.4741
[37]	R. Smarandache, H. Gluesing-Luerssen, J. Rosenthal, Constructions for MDS-convolutional codes, IEEE Trans. Inform. Theory, 47 (2001), 2045–2049. https://doi.org/10.1109/18.930938 doi: 10.1109/18.930938
[38]	The GAP Group, Groups, algorithms, and programming, Version 4.12.2; 2022. Available from: https://www.gap-system.org

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)