Improved binary codes and sequence families from Z/sub 4/-linear codes

1996 ◽  
Vol 42 (5) ◽  
pp. 1582-1587 ◽  
Author(s):  
A.G. Shanbhag ◽  
P. Vijay Kumar ◽  
T. Hellesath
Keyword(s):  
Linear Codes ◽  
Binary Codes

New extremal Type II ℤ4-codes of length 32 obtained from Hadamard matrices

2019 ◽  
Vol 11 (05) ◽  
pp. 1950057
Author(s):  
Sara Ban ◽  
Dean Crnković ◽  
Matteo Mravić ◽  
Sanja Rukavina
Keyword(s):  
Incidence Matrix ◽  
Linear Codes ◽  
Minimum Weight ◽  
Hadamard Matrices ◽  
Binary Codes ◽  
Type Ii ◽  
Hadamard Design ◽  
Binary Linear Codes ◽  
Type Formula ◽  
Hadamard Designs

For every Hadamard design with parameters [Formula: see text]-[Formula: see text] having a skew-symmetric incidence matrix we give a construction of 54 Hadamard designs with parameters [Formula: see text]-[Formula: see text]. Moreover, for the case [Formula: see text] we construct doubly-even self-orthogonal binary linear codes from the corresponding Hadamard matrices of order 32. From these binary codes we construct five new extremal Type II [Formula: see text]-codes of length 32. The constructed codes are the first examples of extremal Type II [Formula: see text]-codes of length 32 and type [Formula: see text], [Formula: see text], whose residue codes have minimum weight 8. Further, correcting the results from the literature we construct 5147 extremal Type II [Formula: see text]-codes of length 32 and type [Formula: see text].

The structure of ℤ2ℤ2s-additive cyclic codes

2018 ◽  
Vol 10 (04) ◽  
pp. 1850048 ◽  
Author(s):  
Ismail Aydogdu ◽  
Taher Abualrub
Keyword(s):  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Optimal Parameter ◽  
Binary Codes ◽  
Binary Linear Codes ◽  
Additive Codes

[Formula: see text]-additive codes for any integer [Formula: see text] are considered as codes over mixed alphabets. They are a generalization of binary linear codes and linear codes over [Formula: see text] In this paper, we are interested in studying [Formula: see text]-additive cyclic codes. We will give the generator polynomials of these codes. We will also give the minimal spanning sets for these codes. We will define separable [Formula: see text]-additive codes and provide conditions on the generator polynomials for a [Formula: see text]-additive cyclic code to be separable. Finally, we present some examples of optimal parameter binary codes obtained as images of [Formula: see text]-additive cyclic codes.

scholarly journals About the Ultimate Efficiency of Interference Binary Codes

2021 ◽  
Author(s):  
Artem Sergeevich Adzhemov ◽  
Nicolay Yurievich Albov
Keyword(s):  
Communication Channel ◽  
Linear Codes ◽  
Binary Codes ◽  
Digital Representation ◽  
Ultimate Efficiency

The digital representation of various signals allows, at the subsequent stages of their transmission, to apply correction codes that provide protection against possible errors arising from the action of interference in the communication channel. At the same time, it is important that, with the required correcting ability, these codes have the maximum possible speed. The article presents the results of calculations for linear codes, showing their really achievable limiting capabilities.

ON THE GENERALISATION OF SIDEL’NIKOV’S THEOREM TO -ARY LINEAR CODES

2019 ◽  
Vol 101 (1) ◽  
pp. 157-162
Author(s):  
YILUN WEI ◽  
BO WU ◽  
QIJIN WANG
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Linear Codes ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Binary Codes ◽  
Normal Distribution Function ◽  
Code Length ◽  
Standard Normal Distribution Function ◽  
Standard Normal

We generalise Sidel’nikov’s theorem from binary codes to $q$-ary codes for $q>2$. Denoting by $A(z)$ the cumulative distribution function attached to the weight distribution of the code and by $\unicode[STIX]{x1D6F7}(z)$ the standard normal distribution function, we show that $|A(z)-\unicode[STIX]{x1D6F7}(z)|$ is bounded above by a term which tends to $0$ when the code length tends to infinity.

scholarly journals Linear, cyclic and constacyclic codes over S4 = F2 + uF2 + u2F2 + u3F2

Filomat ◽  
2014 ◽  
Vol 28 (5) ◽  
pp. 897-906
Author(s):  
Ödemiş Özger ◽  
Ümmü Kara ◽  
Bahattin Yıldız
Keyword(s):  
Linear Codes ◽  
Gray Map ◽  
Binary Codes ◽  
Constacyclic Codes ◽  
Weight Enumerators ◽  
Lee Weight ◽  
Macwilliams Identities

In this work, linear codes over the ring S4 = F2 + uF2 + u2F2 + u3F2 are considered. The Lee weight and gray map for codes over S4 are defined and MacWilliams identities for the complete, the symmetrized and the Lee weight enumerators are obtained. Cyclic and (1 + u2)-constacyclic codes over S4 are studied, as a result of which a substantial number of optimal binary codes of different lengths are obtained as the Gray images of cyclic and constacyclic codes over S4.

Catalyst-Control over Sixfold Stereogenicity

2020 ◽  
Author(s):  
Xingxing Wu ◽  
Reto M. Witzig ◽  
Rodolphe Beaud ◽  
Christian Fischer ◽  
Daniel Häussinger ◽  
...  
Keyword(s):  
Conformational Analysis ◽  
Higher Order ◽  
Binary Codes ◽  
Rotational Barriers ◽  
Molecular Architectures

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

Two-Weight Linear Codes and Their Applications in Secret Sharing

2019 ◽  
Vol 28 (4) ◽  
pp. 706-711
Author(s):  
Yaru Wang ◽  
Fulin Li ◽  
Shixin Zhu
Keyword(s):  
Secret Sharing ◽  
Linear Codes
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