Abstract
Let I be the commutative non-unital ring of order 4 defined by generators and relations.
Alahmadi et al. have classified QSD codes, Type IV codes (QSD codes with even weights) and quasi-Type IV codes (QSD codes with even torsion code) over I up to lengths \(n=6\), and suggested two building-up methods for constructing QSD codes. In this paper, we construct more QSD codes, Type IV codes and quasi-Type IV codes for lengths \(n=7\) and 8, and describe five new variants of the two building-up construction methods. We find that when \(n=8\) there is at least one QSD code with minimun distance 4, which attains the highest minimum distance so far, and we give a generator matrix for the code. We also describe some QSD codes, Type IV codes and quasi-Type IV codes with new weight distributions.
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References
Alahmadi, A., Altassan, A., Basaffar, W., Bonnecaze, A., Shoaib, H., Solé, P.: Quasi Type IV codes over a non-unital ring. Appl. Algebra Eng. Commun. Comput. 32, 217–228 (2021)
Alahmadi, A., Alkathiry, A., Altassan, A., Bonnecaze, A., Shoaib, H., Solé, P.: The build-up construction of quasi self-dual codes over a commutative non-unital ring (2020) https://hal-lirmm.ccsd.cnrs.fr/I2M-2014-/hal-02977595v1
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)
Dougherty, S., Gaborit, P., Harada, M., Munemasa, A., Solé, P.: Type IV self-dual codes over rings. IEEE Trans. Inf. Theory 45(7), 2345–2360 (1999)
Fine, B.: Classification of finite rings of order \(p^{2}\). Math. Mag. 66(4), 248–252 (1993)
Huffman, W., Pless, V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003)
Kim, J.-L.: New extremal self-dual codes of lengths 36, 38, and 58. IEEE Trans. Inf. Theory 47(1), 386–393 (2001)
Kim, J.-L., Lee, Y.: Euclidean and Hermitian self-dual MDS codes over large finite fields. J. Comb. Theory Ser. A 105, 79–95 (2004)
Kim, J.-L., Lee, Y.: An efficient construction of self-dual codes. Bull. Korean Math. Soc. 52(3), 915–923 (2015)
Kim, J.-L., Lee, Y.: Construction of MDS self-dual codes over Galois rings. Des. Codes Cryptogr. 45(2), 247–258 (2007)
Kim, J.-L., Ohk, D.: DNA codes over two noncommutative rings of order four. J. Appl. Math. Comput. (2021) https://doi.org/10.1007/s12190-021-01598-7, arXiv:2102.06981
Pless, V.: Introduction to the Theory of Error-Correcting Codes. John Wiley & Sons Inc, New York (1998)
Acknowledgements
This research (JLK) is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (NRF-2019R1A2C1088676).
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Kim, JL., Roe, Y.G. Construction of quasi self-dual codes over a commutative non-unital ring of order 4. AAECC 35, 393–406 (2024). https://doi.org/10.1007/s00200-022-00553-8
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DOI: https://doi.org/10.1007/s00200-022-00553-8