Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
153 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Additive Complementary Pairs of Codes (2404.13249v3)

Published 20 Apr 2024 in cs.IT and math.IT

Abstract: An additive code is an $\mathbb{F}q$-linear subspace of $\mathbb{F}{qm}n$ over $\mathbb{F}{qm}$, which is not a linear subspace over $\mathbb{F}{qm}$. Linear complementary pairs (LCP) of codes have important roles in cryptography, such as increasing the speed and capacity of digital communication and strengthening security by improving the encryption necessities to resist cryptanalytic attacks. This paper studies an algebraic structure of additive complementary pairs (ACP) of codes over $\mathbb{F}{qm}$. Further, we characterize an ACP of codes in analogous generator matrices and parity check matrices. Additionally, we identify a necessary condition for an ACP of codes. Besides, we present some constructions of an ACP of codes over $\mathbb{F}{qm}$ from LCP codes over $\mathbb{F}{qm}$ and also from an LCP of codes over $\mathbb{F}{q}$. Finally, we study the constacyclic ACP of codes over $\mathbb{F}_{qm}$ and the counting of the constacyclic ACP of codes.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (27)
  1. On Linear Complementary Pairs of Algebraic Geometry Codes over Finite Fields. In arXiv:2311.01008v1 , 2023.
  2. Orthogonal direct sum masking - A smartcard friendly computation paradigm in a code, with built-in protection against side-channel and fault attacks. In Information Security Theory and Practice. Securing the Internet of Things WISTP 2014, Crete, Greece. Proceedings, volume 8501 of Lecture Notes in Computer Science, pages 40–56. Springer, 2014.
  3. Correcting quantum errors with entanglement. Science, 314:436–439, 2006.
  4. A. R. Calderbank, P. W. Shor. Good quantum error-correcting codes exist. Phys. Rev. A, 54(2):1098–1105, 1996.
  5. Constacyclic I⁢FqIsubscriptF𝑞{\rm I\!F}_{q}roman_I roman_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT-linear codes over I⁢FqlIsubscriptFsuperscript𝑞𝑙{\rm I\!F}_{q^{l}}roman_I roman_F start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT end_POSTSUBSCRIPT. Appl. Algebra Eng. Commun. Comput., 26:369-388, 2015.
  6. Complementary dual codes for counter-measures to side-channel attacks. Advances in Mathematics of Communications, 10(1):131–150, 2016.
  7. On linear complementary pairs of codes. IEEE Trans. Inf. Theory, 64(10):6583–6589, 2018.
  8. Linear codes over I⁢FqIsubscriptF𝑞{\rm I\!F}_{q}roman_I roman_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT are equivalent to LCD codes for q>3𝑞3q>3italic_q > 3. IEEE Trans. Inf. Theory, 64(4):3010–3017, 2018.
  9. Euclidean and hermitian LCD MDS codes. Designs, Codes and Cryptography, 86(11):2605–2618, 2018.
  10. On σ𝜎\sigmaitalic_σ-LCD codes. IEEE Trans. Inf. Theory, 65(3):1694–1704, 2019.
  11. On linear complementary pair of n D cyclic codes. IEEE Commun. Lett., 22(12):2404–2406, 2018.
  12. Theory of additive complementary dual codes, constructions and computations. Finite Fileds Appl., 92:102303, 2023.
  13. P. Delsarte. An algebraic approach to the association schemes of coding theory. Philips Res.Rep. Suppl., 1973.
  14. M. F. Ezerman, S. Ling and P. Sole. Additive asymmetric quantum codes. IEEE Transactions on Information Theory, 57: 5536–5550, 2011.
  15. M. Grassl. Bounds on the minimum distance of entanglement-assisted qubit quantum codes. http://eaqecc.codetables.de/qubit.php, (online).
  16. M. Grassl, F. Huber and A. Winter. Entropic Proofs of Singleton Bounds for Quantum Error-Correcting Codes. IEEE Transactions on Information Theory, 68(6): 3942–3952, 2022.
  17. W. C. Huffman. On the theory of I⁢FqIsubscriptF𝑞{\rm I\!F}_{q}roman_I roman_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT-linear I⁢FqtIsubscriptFsuperscript𝑞𝑡{\rm I\!F}_{q^{t}}roman_I roman_F start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT-codes. Advances in Mathematics of Communications, 7: 349–378, 2013.
  18. J. L. Kim and N. Lee. Secret sharing schemes based on additive codes over GF(4). Appl. Algebra Eng. Commun. Comput., 28: 79–97, 2017.
  19. Maximal entanglement entanglement-assisted quantum codes constructed from linear codes. Quantum Information Processing, 14: 165–182, 2015.
  20. J. L. Massey. Linear codes with complementary duals. Discrete Mathematics, 106-107:337–342, 1992.
  21. Complementary dual algebraic geometry codes. IEEE Trans. Inf. Theory, 64(4):2390–2397, 2018.
  22. Linear complementary dual code improvement to strengthen encoded circuit against hardware trojan horses. In IEEE International Symposium on Hardware Oriented Security and Trust, HOST 2015, Washington, DC, USA, 5-7 May 2015, pages 82–87. IEEE Computer Society, 2015.
  23. E. M. Rains Nonbinary quantum codes. IEEE Trans. Inform. Theory, 45:1828-1832, 1999.
  24. LCD and ACP codes over a noncommutative non-unital ring with four elements. Cryptogr. Commun., 14:627-640, 2022.
  25. Additive complementary dual codes over I⁢F4IsubscriptF4{\rm I\!F}_{4}roman_I roman_F start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT. Des., Codes Cryptogr., 91:273-284, 2022.
  26. Additive cyclic complementary dual codes over I⁢F4IsubscriptF4{\rm I\!F}_{4}roman_I roman_F start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT. Finite Fileds Appl., 75:102087, 2022.
  27. M. M. Wilde, T. A. Brun. Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A, 77(064302):1-4, 2008.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets