Separable additive quadratic residue codes over $\mathbb{Z}_2\mathbb{Z}_4$ and their applications
DOI:
https://doi.org/10.13069/jacodesmath.v12i1.353Keywords:
Separable additive quadratic residue codes, Idempotent generators, Self-dual codes, Additive complementary pair codes, Additive $l$-intersection pairs of codesAbstract
This paper examines separable additive quadratic residue codes (QRCs) over $\mathbb{Z}_{2}\mathbb{Z}_{4}$ and their applications. The idempotent generators of these codes are obtained. Further, the properties of separable additive QRCs over $\mathbb{Z}_{2}\mathbb{Z}_{4}$ are studied including their idempotent generators. As applications, these codes are used to construct self-dual, self-orthogonal, additive complementary pair (ACP) codes, additive complementary dual (ACD) codes, and additive $l$-intersection pairs of codes over $\mathbb{Z}_{2}\mathbb{Z}_{4}$.
Received: 5 October 2024 | Accepted: 8 November 2024Downloads
References
T. Abualrub, I. Siap, and N. Aydin, $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes, IEEE Trans. Inform. Theory 60(3) (2014) 1508–1514.
E. F. Assmus Jr. and J. D. Key, Affine and projective planes, Discrete Math. 83(2–3) (1990) 161–187.
S. Bhowmick, A. Fotue-Tabue, and J. Pal, On the $\ell$-DLIPs of codes over finite principal ideal rings, Discrete Math. 347(4) (2024) 113853.
N. Benbelkacem, J. Borges, S. T. Dougherty, and C. Fernández-Córdoba, On $\mathbb{Z}_2\mathbb{Z}_4$-additive complementary dual codes and related LCD codes, Finite Fields Appl. 62 (2020) 101622.
J. Borges, C. Fernández-Córdoba, S. T. Dougherty, and R. Ten-Valls, Binary images of $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes, IEEE Trans. Inform. Theory 64(12) (2018) 7551–7556.
J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rifà, and M. Villanueva, $\mathbb{Z}_2\mathbb{Z}_4$-linear codes: Generator matrices and duality, Des. Codes Cryptogr. 54(2) (2010) 167–179.
J. Borges, C. Fernández-Córdoba, and R. Ten-Valls, $\mathbb{Z}_2$-double cyclic codes, Des. Codes Cryptogr. 86(3) (2018) 463–479.
H. Ghosh, P. K. Maurya, and S. Bagchi, Secret sharing scheme: Based on LCD code, in 2022 International Interdisciplinary Conference on Mathematics, Engineering and Science (MESIICON), Durgapur, India (2022) 1–4.
K. Guenda, T. A. Gulliver, S. Jitman, and S. Thipworawimon, Linear $\ell$-intersection pairs of codes and their applications, Des. Codes Cryptogr. 88(1) (2020) 133–152.
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge (2003).
A. S. Karbaski, T. Abualrub, and S. T. Dougherty, Double quadratic residue codes and self-dual double cyclic codes, Appl. Algebra Engrg. Comm. Comput. 33(2) (2022) 91–115.
X. T. Ngo, S. Bhasin, J.-L. Danger, S. Guilley, and Z. Najm, Linear complementary dual code improvement to strengthen encoded circuit against hardware Trojan horses, in 2015 IEEE International Symposium on Hardware Oriented Security and Trust (HOST), Washington, DC, USA (2015) 82–87.
V. Pless and Z. Qian, Cyclic codes and quadratic residue codes over $\mathbb{Z}_4$, IEEE Trans. Inform. Theory 42(5) (1996) 1594–1600.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam (1977).
E. Prange, Some cyclic error-correcting codes with simple decoding algorithms, Air Force Cambridge Research Center, Bedford, MA, USA, AFCRC-TN-58-156 (1958).
J. Rifà and L. Ronquillo, Product perfect $\mathbb{Z}_2\mathbb{Z}_4$-linear codes in steganography, in 2010 International Symposium on Information Theory & Its Applications (ISITA), Taichung, Taiwan (2010) 696–701.
Z.-X. Wan, Quaternary Codes, World Scientific, Singapore (1997).
