Dual transform and projective self-dual codes

Iliya Bouyukliev; Stefka Bouyuklieva

doi:10.3934/amc.2023032

Article Contents

2024, Volume 18, Issue 2: 328-341. Doi: 10.3934/amc.2023032

This issue Previous Article Various structures of cyclic codes over the non-Frobenius ring $\mathbb{F}_p[u, v] /\left\langle u^2, v^2, u v, v u\right\rangle$ Next Article PMNS for cryptography: A guided tour

Dual transform and projective self-dual codes

Iliya Bouyukliev^1, and
Stefka Bouyuklieva^2, ,

1.
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, P.O. Box 323, Veliko Tarnovo, Bulgaria
2.
Faculty of Mathematics and Informatics, St. Cyril and St. Methodius University of Veliko Tarnovo, Bulgaria

^*Corresponding author: Stefka Bouyuklieva
^*Corresponding author: Stefka Bouyuklieva

Received: April 2023

Revised: August 2023

Early access: August 2023

Published: April 2024

Both authors are supported by [the Bulgarian National Science Fund under Contract No KP-06-H62/2/13.12.2022].

Abstract / Introduction Full Text(HTML) Figure(0) / Table(1) Related Papers Cited by

Abstract

We present and analyze the relation between different types of linear codes through the (projective) dual transform. The considered codes are represented by a generator matrix or a characteristic vector, and the dual transform is defined in these terms. This allows us to extend its use to study codes with special properties, which we call self-polar, and their relation to Boolean functions. The self-polar codes are a special class of projective self-dual codes and are closely connected with self-dual and anti-self-dual bent functions. We also give some computational results for self-dual bent functions in eight variables.

Keywords:

Mathematics Subject Classification: Primary: 94B05, 05B25, 06E30.

Citation:

Full Text(HTML)

Table 1. Orders of the automorphism groups of the codes $ C(f) $ for the constructed self-dual bent functions $ f $

$ \|Aut(C)\| $	$ \sharp $ codes	$ \|Aut(C)\| $	$ \sharp $ codes	$ \|Aut(C)\| $	$ \sharp $ codes	$ \|Aut(C)\| $	$ \sharp $ codes
1032192	1	98304	2	144	2	96	12
589824	1	24576	2	768	6	48	13
786432	1	49152	4	1536	8	24	29
344064	1	3840	1	240	1	72	3
393216	2	1344	2	384	5	12	39
294912	1	3072	7	192	2	30	1
196608	2	6144	3	42	1	9	1
86016	1	9216	1	18	1	6	50
51840	1	1920	2	36	4	3	23

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	E. F. Assmus and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, Vol. 103, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781316529836.
[2]	I. G. Bouyukliev, Classification of Griesmer codes and dual transform, Discrete Mathematics, 309 (2009), 4049-4068. doi: 10.1016/j.disc.2008.12.002.
[3]	S. Bouyuklieva and I. Bouyukliev, Dual transform through characteristic vectors, Proceedings of the International Workshop OCRT, Sofia, Bulgaria, 2017, 43-48.
[4]	I. Bouyukliev, S. Bouyuklieva and S. Dodunekov, On binary self-complementary $[120, 9, 56]$ codes having an automorphism of order 3 and associated SDP designs, Probl. Inf. Transm., 43 (2007), 89-96. doi: 10.1134/S0032946007010020.
[5]	I. Bouyukliev, S. Bouyuklieva and S. Kurz, Computer classification of linear codes, IEEE Trans. Inform. Theory, 67 (2021), 7807-7814. doi: 10.1109/TIT.2021.3114280.
[6]	I. Bouyukliev, S. Bouyuklieva, T. Maruta and P. Piperkov, Characteristic vector and weight distribution of a linear code, Cryptogr. Commun., 13 (2021), 263-282. doi: 10.1007/s12095-020-00458-8.
[7]	I. Bouyukliev, S. Bouyuklieva and M. Pashinska-Gadzheva, On some families of codes related to the even linear codes meeting the Grey-Rankin bound, Mathematics, 10 (2022), 4588. doi: 10.3390/math10234588.
[8]	A. E. Brouwer, P. J. Cameron, W. H. Haemers and D. A. Preece, Self-dual, not self-polar, Discrete Mathematics, 306 (2006), 3051-3053. doi: 10.1016/j.disc.2004.11.027.
[9]	A. E. Brouwer and M. van Eupen, The correspondence between projective codes and 2-weight codes, Des., Codes and Crypt., 11 (1997), 262-266. doi: 10.1023/A:1008294128110.
[10]	R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122. doi: 10.1112/blms/18.2.97.
[11]	C. Carlet, Boolean functions for cryptography and error correcting codes, Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Cambridge University Press, Cambridge, 2010,257-397. doi: 10.1017/CBO9780511780448.011.
[12]	C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Designs, Codes and Cryptography, 15 (1998), 125-156. doi: 10.1023/A:1008344232130.
[13]	C. Carlet, L. E. Danielsen, M. G. Parker and P. Solé, Self-dual bent functions, Int. J. Inform. Coding Theory, 1 (2010), 384-399. doi: 10.1504/IJICOT.2010.032864.
[14]	P. Delsarte, Weights of linear codes and strongly regular normed spaces, Discrete Mathematics, 3 (1972), 47-64. doi: 10.1016/0012-365X(72)90024-6.
[15]	C. Ding, Linear codes from some 2-designs, IEEE Trans. Inform. Theory, 61 (2015), 3265-3275. doi: 10.1109/TIT.2015.2420118.
[16]	C. Ding, A construction of binary linear codes from Boolean functions, Discrete Mathematics, 339 (2016), 2288-2303. doi: 10.1016/j.disc.2016.03.029.
[17]	S. Dodunekov and J. Simonis, Codes and projective multisets, Electron. J. Combin., 5 (1998), Research Paper 37, 23 pp. doi: 10.37236/1375.
[18]	Y. Edel and A. Pott, On the equivalence of nonlinear functions, Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, IOS Press, 2009, 87-103.
[19]	R. Hill, Caps and codes, Discrete Mathematics, 22 (1978), 111-137. doi: 10.1016/0012-365X(78)90120-6.
[20]	W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511807077.
[21]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier, North-Holland, Amsterdam, 1977.
[22]	S. Mesnager, Bent Functions: Fundamentals and Results, Springer, 2016. doi: 10.1007/978-3-319-32595-8.