16 research outputs found
New Hermitian self-dual MDS or near-MDS codes over finite fields
A linear code over a finite field is called Hermitian self-dual if the code is self-dual under the Hermitian inner-product. The Hermitian self-dual code is called MDS or near-MDS if the code attains or almost attains the Singleton bound. In this paper we construct new Hermitian self-dual MDS or near-MDS codes over and of length up to 14
Research on Hermitian self-dual codes, GRS codes and EGRS codes
MDS self-dual codes have nice algebraic structures, theoretical significance
and practical implications. In this paper, we present three classes of
-ary Hermitian self-dual (extended) generalized Reed-Solomon codes with
different code locators. Combining the results in Ball et al. (Designs, Codes
and Cryptography, 89: 811-821, 2021), we show that if the code locators do not
contain zero, -ary Hermitian self-dual (extended) GRS codes of length
does not exist. Under certain conditions, we prove Conjecture
3.7 and Conjecture 3.13 proposed by Guo and Li et al. (IEEE Communications
Letters, 25(4): 1062-1065, 2021).Comment: 18 page
On the classification of Hermitian self-dual additive codes over GF(9)
Additive codes over GF(9) that are self-dual with respect to the Hermitian
trace inner product have a natural application in quantum information theory,
where they correspond to ternary quantum error-correcting codes. However, these
codes have so far received far less interest from coding theorists than
self-dual additive codes over GF(4), which correspond to binary quantum codes.
Self-dual additive codes over GF(9) have been classified up to length 8, and in
this paper we extend the complete classification to codes of length 9 and 10.
The classification is obtained by using a new algorithm that combines two graph
representations of self-dual additive codes. The search space is first reduced
by the fact that every code can be mapped to a weighted graph, and a different
graph is then introduced that transforms the problem of code equivalence into a
problem of graph isomorphism. By an extension technique, we are able to
classify all optimal codes of length 11 and 12. There are 56,005,876
(11,3^11,5) codes and 6493 (12,3^12,6) codes. We also find the smallest codes
with trivial automorphism group.Comment: 12 pages, 6 figure
and
The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(pe, l) (including Zpe). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings GF (2e, l) of length n = 2l for any a ≥ 1 and l ≥ 2. Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem