4 research outputs found
Around Pelikan's conjecture on very odd sequences
Very odd sequences were introduced in 1973 by J. Pelikan who conjectured that
there were none of length >=5. This conjecture was disproved by MacWilliams and
Odlyzko in 1977 who proved there are in fact many very odd sequences. We give
connections of these sequences with duadic codes, cyclic difference sets,
levels (Stufen) of cyclotomic fields, and derive some new asymptotic results on
their lengths and on S(n), which denotes the number of very odd sequences of
length n.Comment: 21 pages, two tables. Revised version with improved presentation and
correction of some typos and minor errors that will appear in Manuscripta
Mathematic
Properties and classifications of certain LCD codes.
A linear code is called a linear complementary dual code (LCD code) if holds. LCD codes have many applications in cryptography, communication systems, data storage, and quantum coding theory. In this dissertation we show that a necessary and sufficient condition for a cyclic code over of odd length to be an LCD code is that where is a self-reciprocal polynomial in which is also in our paper \cite{GK1}. We then extend this result and provide a necessary and sufficient condition for a cyclic code of length over a finite chain ring R=\big(R,\m=(\gamma),\kappa=R/\m \big) with to be an LCD code. In \cite{DKOSS} a linear programming bound for LCD codes and the definition for for binary LCD -codes are provided. Thus, in a different direction, we find the formula for which appears in \cite{GK2}. In 2020, Pang et al. defined binary codes with biggest minimal distance, which meets the Griesmer bound \cite{Pang}. We give a correction to and provide a different proof for \cite[Theorem 4.2]{Pang}, provide a different proof for \cite[Theorem 4.3]{Pang}, examine properties of LCD ternary codes, and extend some results found in \cite{Harada} for any which is a power of an odd prime
Duadic codes over F2 + uF2
10.1007/s002000100079Applicable Algebra in Engineering, Communications and Computing125365-379AAEC