scholarly journals On the binary codes with parameters of triply-shortened 1-perfect codes

2011 ◽  
Vol 64 (3) ◽  
pp. 275-283 ◽  
Author(s):  
Denis S. Krotov
Keyword(s):  
10.37236/3220 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Joaquim Borges ◽  
Ivan Yu. Mogilnykh ◽  
Josep Rifà ◽  
Faina I. Solov'eva

The paper proves that there exists an exponential number of nonequivalent propelinear extended perfect binary codes of length growing to infinity. Specifically, it is proved that all transitive extended perfect binary codes found by Potapov (2007) are propelinear. All such codes have small rank, which is one more than the rank of the extended Hamming code of the same length. We investigate the properties of these codes and show that any of them has a normalized propelinear representation.


2018 ◽  
Vol 18 (5) ◽  
pp. 63-76
Author(s):  
Alexander A. Davydov ◽  
Stefano Marcugini ◽  
Fernanda Pambianco

Abstract Binary codes created by doubling construction, including quasi-perfect ones with distance d = 4, are investigated. All [17·2r−6, 17·2r−6 − r, 4] quasi-perfect codes are classified. Weight spectrum of the codes dual to quasi-perfect ones with d = 4 is obtained. The automorphism group Aut(C) of codes obtained by doubling construction is studied. A subgroup of Aut(C) is described and it is proved that the subgroup coincides with Aut(C) if the starting matrix of doubling construction has an odd number of columns. (It happens for all quasi-perfect codes with d = 4 except for Hamming one.) The properness and t-properness for error detection of codes obtained by doubling construction are considered.


2019 ◽  
Vol 11 (02) ◽  
pp. 1950018
Author(s):  
Vinod Tyagi ◽  
Tarun Lata
Keyword(s):  

Etzion [T. Etzion, Perfect byte correcting codes, IEEE Trans. Inform. Theory 44 (1998) 3140–3146.] has classified byte error correcting binary codes into five different categories with respect to the size of the byte. If a code is partitioned into [Formula: see text] equal byte of size say [Formula: see text] then [Formula: see text] or the size of bytes is [Formula: see text]. Alternatively if bytes are of different size say, [Formula: see text] then [Formula: see text]. The result was further modified by Tyagi and Sethi [V. Tyagi and A. Sethi, [Formula: see text]-Byte correcting perfect codes, Asian-Eur. J. Math. 7(1) (2014) 1–8.] and the classification of bytes was given with respect to size of the byte as well as length of the burst. We call such codes as [Formula: see text]-byte correcting perfect codes. Our aim in this paper is to find the possibilities for the existence of [Formula: see text]-byte correcting non-binary perfect codes.


2020 ◽  
Author(s):  
Xingxing Wu ◽  
Reto M. Witzig ◽  
Rodolphe Beaud ◽  
Christian Fischer ◽  
Daniel Häussinger ◽  
...  

Governing higher-order stereogenicity is a long-standing goal in stereoselective catalysis, because it allows to achieve selectivity for more than a twofold number of stereoisomers per stereogenic unit. Current methods warrant control over the power of two stereoisomers and the configurations are routinely assigned using the descriptors ( R ) and ( S ), or related binary codes. In contrast, conformational analysis ranges beyond this dualistic treatment of stereoisomerism, which constitutes an unmet challenge for catalyst stereocontrolled processes. Herein, we now report that sixfold stereogenicity can be governed by stereoselective catalysis. By controlling a configurationally stable stereogenic axis with six large rotational barriers, a catalytic [2+2+2]-cyclotrimerization selectively governs the formation of one out of six stereoisomers with up to 0:0:2:98:0:0 stereocontrol. The underpinnings of conformational analysis and stereoselective catalysis are thereby conceptually reunited. Novel molecular architectures featuring distinct chemical topologies and unexplored chemical designs are anticipated from catalystcontrol over higher-order stereogenicities


2021 ◽  
Vol 40 (2) ◽  
Author(s):  
Giselle Strey ◽  
João E. Strapasson ◽  
Sueli I. R. Costa

2021 ◽  
Vol 344 (3) ◽  
pp. 112261
Author(s):  
Zihui Liu
Keyword(s):  

2008 ◽  
Vol 54 (4) ◽  
pp. 1760-1765 ◽  
Author(s):  
Denis S. Krotov ◽  
Sergey V. Avgustinovich

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