1,445 research outputs found
Euclidean and Hermitian LCD MDS codes
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) have been of
much interest from many researchers due to their theoretical significant and
practical implications. However, little work has been done on LCD MDS codes. In
particular, determining the existence of -ary LCD MDS codes for
various lengths and dimensions is a basic and interesting problem. In
this paper, we firstly study the problem of the existence of -ary
LCD MDS codes and completely solve it for the Euclidean case. More
specifically, we show that for there exists a -ary Euclidean
LCD MDS code, where , or, , and . 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
Entanglement-assisted quantum error-correcting codes over arbitrary finite fields
We prove that the known formulae for computing the optimal number of maximally entangled pairs required for entanglement-assisted quantum error-correcting codes (EAQECCs) over the binary field hold for codes over arbitrary finite fields as well. We also give a Gilbert-Varshamov bound for EAQECCs and constructions of EAQECCs coming from punctured self-orthogonal linear codes which are valid for any finite field.Supported by the Spanish Ministry of Economy/FEDER: grants MTM2015-65764-C3-1-P, MTM2015-65764-C3-2-P, MTM2015-69138-REDT and RYC-2016-20208 (AEI/FSE/UE), the University Jaume I: grant UJI-B2018-10, Spanish Junta de CyL: grant VA166G18, and JSPS Grant No. 17K06419
- …