Designs, Codes and Cryptography

, Volume 86, Issue 11, pp 2605–2618

# Euclidean and Hermitian LCD MDS codes

• Claude Carlet
• Sihem Mesnager
• Chunming Tang
• Yanfeng Qi
Article

## Abstract

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) are of much interest from many viewpoints due to their theoretical and practical properties. However, little work has been done on LCD MDS codes. In particular, determining the existence of q-ary [nk] LCD MDS codes for various lengths n and dimensions k is a basic and interesting problem. In this paper, we firstly study the problem of the existence of q-ary [nk] LCD MDS codes and solve it for the Euclidean case. More specifically, we show that for $$q>3$$ there exists a q-ary [nk] Euclidean LCD MDS code, where $$0\le k \le n\le q+1$$, or, $$q=2^{m}$$, $$n=q+2$$ and $$k= 3 \text { or } q-1$$. Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes.

## Keywords

Linear codes MDS codes Linear complementary dual Self-dual codes Self-orthogonal codes

94B05 94B65

## Notes

### Acknowledgements

The authors thank the Assoc. Edit. and the anonymous reviewers for their valuable comments which have highly improved the manuscript. This work was supported by SECODE project and the National Natural Science Foundation of China (Grant Nos. 11401480, 11531002, 11701129). The work of C. Carlet was partly supported by Bergen Research Foundation. C. Tang also acknowledges support from 14E013, CXTD2014-4 and the Meritocracy Research Funds of China West Normal University. Y. Qi also acknowledges support from Zhejiang provincial Natural Science Foundation of China (LQ17A010008, LQ16A010005).

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