AN ALGORITHM FOR COMPUTING THE WEIGHT DISTRIBUTION OF A LINEAR CODE OVER A COMPOSITE FINITE FIELD WITH CHARACTERISTIC 2

2019 ◽  
Author(s):  
Paskal PIPERKOV ◽  
Iliya BOUYUKLIEV ◽  
Stefka BOUYUKLIEVA
Keyword(s):  
Finite Field ◽  
Weight Distribution ◽  
Linear Code ◽  
Characteristic 2

On some punctured codes of ℤq-Simplex codes

2021 ◽  
pp. 2250012
Author(s):  
J. Prabu ◽  
J. Mahalakshmi ◽  
C. Durairajan ◽  
S. Santhakumar
Keyword(s):  
Prime Divisor ◽  
Weight Distribution ◽  
Linear Code ◽  
Prime Power ◽  
Punctured Codes ◽  
New Codes ◽  
Simplex Codes ◽  
Simplex Code ◽  
Partial Weight ◽  
Unit Formula

In this paper, we have constructed some new codes from [Formula: see text]-Simplex code called unit [Formula: see text]-Simplex code. In particular, we find the parameters of these codes and have proved that it is a [Formula: see text] [Formula: see text]-linear code, where [Formula: see text] and [Formula: see text] is a smallest prime divisor of [Formula: see text]. When rank [Formula: see text] and [Formula: see text] is a prime power, we have given the weight distribution of unit [Formula: see text]-Simplex code. For the rank [Formula: see text] we obtain the partial weight distribution of unit [Formula: see text]-Simplex code when [Formula: see text] is a prime power. Further, we derive the weight distribution of unit [Formula: see text]-Simplex code for the rank [Formula: see text] [Formula: see text].

Maximal T-spaces of free associative algebras over a finite field

2018 ◽  
Vol 17 (04) ◽  
pp. 1850064
Author(s):  
C. Bekh-Ochir ◽  
S. A. Rankin
Keyword(s):  
Finite Field ◽  
Positive Characteristic ◽  
Associative Algebras ◽  
Prime Field ◽  
Characteristic 2 ◽  
Infinite Set

In earlier work, it was established that for any finite field [Formula: see text] and any nonempty set [Formula: see text], the free associative (nonunitary) [Formula: see text]-algebra on [Formula: see text], denoted by [Formula: see text], had infinitely many maximal [Formula: see text]-spaces, but exactly two maximal [Formula: see text]-ideals (each of which was shown to be a maximal [Formula: see text]-space). This raises the interesting question as to whether or not the maximal [Formula: see text]-spaces can be classified. However, aside from the two maximal [Formula: see text]-ideals, no examples of maximal [Formula: see text]-spaces of [Formula: see text] have been identified to this point. This paper presents, for each finite field [Formula: see text], an infinite set of proper [Formula: see text]-spaces [Formula: see text] of [Formula: see text], none of which is a [Formula: see text]-ideal. It is proven that for any distinct integers [Formula: see text], [Formula: see text]. Furthermore, it is proven that for the prime field [Formula: see text], [Formula: see text] any prime, [Formula: see text] is a maximal [Formula: see text]-space of [Formula: see text]. We conjecture that for any finite field [Formula: see text] of positive characteristic different from 2 and each integer [Formula: see text], [Formula: see text] is a maximal [Formula: see text]-space of [Formula: see text]. In characteristic 2, the situation is slightly different and we provide different candidates for maximal [Formula: see text]-spaces.

On the ℤq-Simplex codes and its weight distribution for dimension 2

2015 ◽  
Vol 07 (03) ◽  
pp. 1550030
Author(s):  
C. Durairajan ◽  
J. Mahalakshmi ◽  
P. Chella Pandian
Keyword(s):  
Weight Distribution ◽  
Linear Code ◽  
Prime Power ◽  
Dimension 2 ◽  
New Codes ◽  
Simplex Codes ◽  
Order Element

In this paper, we have defined ℤq-linear code and constructed some new codes. In particular, we have introduced the concept of ℤq-Simplex codes and proved that it is a [Formula: see text]-linear code for any integer q ≥ 2 and k ≥ 3 where p is the least order element in ℤq. We have given the weight distribution of ℤq-Simplex codes of dimension 2 when q is a prime power and when q is a product of distinct primes.

scholarly journals The autocorrelation of the Möbius function and Chowla's conjecture for the rational function field in characteristic 2

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140311 ◽  
Author(s):  
Dan Carmon
Keyword(s):  
Finite Field ◽  
Rational Function ◽  
Function Field ◽  
Möbius Function ◽  
Rational Function Field ◽  
Mobius Function ◽  
Characteristic 2

We prove a function field version of Chowla's conjecture on the autocorrelation of the Möbius function in the limit of a large finite field of characteristic 2, extending previous work in odd characteristic.

On the unitary units of the group algebra 𝔽2kQ16

2015 ◽  
Vol 08 (01) ◽  
pp. 1550013 ◽  
Author(s):  
Zahid Raza ◽  
Riasat Ali
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Quaternion Group ◽  
Characteristic 2

In this note, the structure of unitary unit groups V*(𝔽2kQ16) is investigated, where Q16, is quaternion group of order 16, and 𝔽2k is any finite field of characteristic 2, with 2k elements. In particular, we give the center Z(V*(𝔽2kQ16)) of unitary units subgroup V*(𝔽2kQ16) of group algebra 𝔽2kQ16. The structure of the unitary unit subgroup V*(𝔽2kQ16) is described with the help of the Z(V*(𝔽2kQ16)).

scholarly journals Infinite Families of 2-Designs from a Class of Linear Codes Related to Dembowski-Ostrom Functions

2021 ◽  
pp. 1-15
Author(s):  
Rong Wang ◽  
Xiaoni Du ◽  
Cuiling Fan ◽  
Zhihua Niu
Keyword(s):  
Finite Field ◽  
Weight Distribution ◽  
Coding Theory ◽  
Exponential Sums ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Research Interest ◽  
Combinatorial Formula ◽  
Text Design ◽  
Fixed Weight

Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial [Formula: see text]-designs have attracted lots of research interest for decades. The interplay between coding theory and [Formula: see text]-designs started many years ago. It is generally known that [Formula: see text]-designs can be used to derive linear codes over any finite field, and that the supports of all codewords with a fixed weight in a code also may hold a [Formula: see text]-design. In this paper, we first construct a class of linear codes from cyclic codes related to Dembowski-Ostrom functions. By using exponential sums, we then determine the weight distribution of the linear codes. Finally, we obtain infinite families of [Formula: see text]-designs from the supports of all codewords with a fixed weight in these codes. Furthermore, the parameters of [Formula: see text]-designs are calculated explicitly.

scholarly journals Weight Distribution of Some Codewords of 3-ary Linear Code over GF(27)‎

2020 ◽  
Vol 31 (4) ◽  
pp. 101
Author(s):  
Maha Majeed Ibrahim ◽  
Emad Bakr Al-Zangana
Keyword(s):  
Weight Distribution ◽  
Linear Code ◽  
Linear Codes ◽  
Generator Matrix ◽  
Weight Distributions

This paper is devoted to introduce the structure of the p-ary linear codes C(n,q) of points and lines of PG(n,q),q=p^h prime. When p=3, the linear code C(2,27) is given with its generator matrix and also, some of weight distributions are calculated.

ON THE UNITARY UNITS OF THE GROUP ALGEBRA 𝔽2mM16

2013 ◽  
Vol 12 (08) ◽  
pp. 1350059 ◽  
Author(s):  
ZAHID RAZA ◽  
MAQSOOD AHMAD
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Modular Group ◽  
Characteristic 2

In this note, we have given the center Z(V*(𝔽2mM16)) of unitary units subgroup V*(𝔽2mM16) of group algebra 𝔽2mM16, where M16 = 〈x, y | x8 = y2 = 1, xy = yx5〉 is the Modular group of order 16 and 𝔽2m is any finite field of characteristic 2, with 2m elements. The structure of the unitary unit subgroup V*(𝔽2mM16) of the group algebra 𝔽2mM16, is also described, see Theorem 3.1.

The analog of the Erdös distance problem in finite fields

2017 ◽  
Vol 13 (09) ◽  
pp. 2319-2333
Author(s):  
S. D. Adhikari ◽  
Anirban Mukhopadhyay ◽  
M. Ram Murty
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Kloosterman Sums ◽  
Vector Spaces ◽  
Characteristic 2 ◽  
Distance Problem ◽  
Erdős Distance Problem ◽  
Erdos Distance Problem

In this paper, we give a proof of the result of Iosevich and Rudnev [Erdös distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc. 359 (2007) 6127–6142] on the analog of the Erdös–Falconer distance problem in the case of a finite field of characteristic [Formula: see text], where [Formula: see text] is an odd prime, without using estimates for Kloosterman sums. We also address the case of characteristic 2.

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