Linear codes over 𝔽4R and their MacWilliams identity

2020 ◽  
Vol 12 (06) ◽  
pp. 2050085
Author(s):  
Nasreddine Benbelkacem ◽  
Martianus Frederic Ezerman ◽  
Taher Abualrub
Keyword(s):  
Commutative Ring ◽  
Linear Codes ◽  
Inner Product ◽  
Dual Codes ◽  
Macwilliams Identity ◽  
Generator Matrices

Let [Formula: see text] be the field of four elements. We denote by [Formula: see text] the commutative ring, with [Formula: see text] elements, [Formula: see text] with [Formula: see text]. This work defines linear codes over the ring of mixed alphabets [Formula: see text] as well as their dual codes under a nondegenerate inner product. We then derive the systematic form of the respective generator matrices of the codes and their dual codes. We wrap the paper up by proving the MacWilliams identity for linear codes over [Formula: see text].

scholarly journals DUAL CODE AND MACWILLIAMS IDENTITY ON ADDITIVE CODE

2019 ◽  
Vol 1 (1) ◽  
Author(s):  
Miftah Yuliati ◽  
Sri Wahyuni ◽  
Indah Emilia Wijayanti
Keyword(s):  
Weight Distribution ◽  
Linear Code ◽  
Hamming Distance ◽  
Finite Abelian Group ◽  
Dual Code ◽  
Inner Product ◽  
Dual Codes ◽  
Macwilliams Identity ◽  
Homogeneous Weight ◽  
Distance Hamming

Additive code is a generalization of linear code. It is defined as subgroup of a finite Abelian group. The definitions of Hamming distance, Hamming weight, weight distribution, and homogeneous weight distribution in additive code are similar with the definitions in linear code. Different with linear code where the dual code is defined using inner product, additive code using theories in group to define its dual code because in group theory we do not have term of inner product. So, by this thesis, the definitions of dual code in additive code will be discussed. Then, this thesis discuss about a familiar theorem in dual code theory, that is MacWilliams Identity. Next, this thesis discuss about how to proof of MacWilliams Identity on adiitive code using dual codes which are defined.

scholarly journals Maximal Arcs, Codes, and New Links Between Projective Planes of Order 16

10.37236/9008 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Mustafa Gezek ◽  
Rudi Mathon ◽  
Vladimir D. Tonchev
Keyword(s):  
Minimum Distance ◽  
Linear Codes ◽  
Projective Planes ◽  
Upper And Lower Bounds ◽  
Dual Codes ◽  
Binary Linear Codes ◽  
Incidence Matrices ◽  
Maximal Arc ◽  
Maximal Arcs ◽  
Even Order

In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence matrices are derived. A lower bound on the minimum distance of the dual codes is proved, and it is shown that the bound is achieved if and only if the related maximal arc contains a hyperoval of the plane. The  binary linear codes of length 52 spanned by the incidence matrices of 2-$(52,4,1)$ designs associated with previously known and some newly found maximal arcs of degree 4 in projective planes of order 16 are analyzed and classified up to equivalence. The classification shows that some designs associated with maximal arcs in nonisomorphic planes generate equivalent codes. This phenomenon establishes new links between several of the known planes. A conjecture concerning the codes of maximal arcs in $PG(2,2^m)$ is formulated.

Structure of repeated-root constacyclic codes of length 8ℓmpn

2019 ◽  
Vol 12 (04) ◽  
pp. 1950050
Author(s):  
Saroj Rani
Keyword(s):  
Finite Field ◽  
Linear Codes ◽  
Important Class ◽  
Constacyclic Codes ◽  
Dual Codes ◽  
Negacyclic Codes ◽  
Positive Integers

Constacyclic codes form an important class of linear codes which is remarkable generalization of cyclic and negacyclic codes. In this paper, we assume that [Formula: see text] is the finite field of order [Formula: see text] where [Formula: see text] is a power of the prime [Formula: see text] and [Formula: see text] are distinct odd primes, and [Formula: see text] are positive integers. We determine generator polynomials of all constacyclic codes of length [Formula: see text] over the finite field [Formula: see text] We also determine their dual codes.

THE EQUIVALENT IDENTITIES OF THE MACWILLIAMS IDENTITIES FOR LINEAR CODES

2015 ◽  
Vol 76 (1) ◽  
Author(s):  
Bao Xiaomin
Keyword(s):  
Linear Codes ◽  
Macwilliams Identity ◽  
Macwilliams Identities

We use derivatives to prove the equivalences between MacWilliams identity and its four equivalent forms, and present new interpretations for the four equivalent forms.

$${{{\mathbb Z}_2}{{\mathbb Z}_4}}$$ -linear codes: generator matrices and duality

2009 ◽  
Vol 54 (2) ◽  
pp. 167-179 ◽  
Author(s):  
J. Borges ◽  
C. Fernández-Córdoba ◽  
J. Pujol ◽  
J. Rifà ◽  
M. Villanueva
Keyword(s):  
Linear Codes ◽  
Generator Matrices

scholarly journals Isodual and self-dual codes from graphs

2021 ◽  
Vol 32 (1) ◽  
pp. 49-64
Author(s):  
S. Mallik ◽  
◽  
B. Yildiz ◽  
Keyword(s):  
Minimum Distance ◽  
Linear Codes ◽  
Type I ◽  
Type Ii ◽  
Dual Codes ◽  
Generator Matrix ◽  
Combinatorial Interpretation ◽  
Binary Linear Codes ◽  
Graph Theoretic ◽  
Graph Classes

Binary linear codes are constructed from graphs, in particular, by the generator matrix [In|A] where A is the adjacency matrix of a graph on n vertices. A combinatorial interpretation of the minimum distance of such codes is given. We also present graph theoretic conditions for such linear codes to be Type I and Type II self-dual. Several examples of binary linear codes produced by well-known graph classes are given.

scholarly journals Type IV codes over a non-unital ring

2021 ◽  
pp. 2250142
Author(s):  
Adel Alahmadi ◽  
Alaa Altassan ◽  
Widyan Basaffar ◽  
Hatoon Shoaib ◽  
Alexis Bonnecaze ◽  
...  
Keyword(s):  
Local Ring ◽  
Algebraic Structure ◽  
Invariant Theory ◽  
Linear Codes ◽  
Hamming Weight ◽  
Dual Codes ◽  
Weight Enumerators ◽  
Type Iv ◽  
Commutative Local Ring ◽  
Torsion Codes

There is a special local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by [Formula: see text] We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over [Formula: see text] and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

scholarly journals On Group Codes Over Elementary Abelian Groups

2003 ◽  
Vol 8 (2) ◽  
pp. 145
Author(s):  
Adnan A. Zain
Keyword(s):  
Abelian Group ◽  
Finite Fields ◽  
Linear Codes ◽  
Abelian Groups ◽  
Parity Check ◽  
Group Codes ◽  
New Codes ◽  
Generator Matrices

For group codes over elementary Abelian groups we present definitions of the generator and the parity check matrices, which are matrices over the ring of endomorphism of the group. We also lift the theorem that relates the parity check and the generator matrices of linear codes over finite fields to group codes over elementary Abelian groups. Some new codes that are MDS, self-dual, and cyclic over the Abelian group with four elements are given.

Characterization of some minimal codes for secret sharing

2019 ◽  
Vol 12 (02) ◽  
pp. 1950026
Author(s):  
Sassia Makhlouf ◽  
Lemnouar Noui
Keyword(s):  
Secret Sharing ◽  
Linear Codes ◽  
Mds Codes ◽  
Secret Sharing Schemes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dual Codes ◽  
Necessary And Sufficient ◽  
Minimal Codes

Recently, several authors used linear codes to construct secret sharing schemes. It is known that if each nonzero codeword of a code [Formula: see text] is minimal, then the dual code [Formula: see text] is suitable for secret sharing. To seek such codes Ashikhmin–Barg give a sufficient condition from weights; in [Formula: see text] code [Formula: see text], let [Formula: see text] and [Formula: see text] be the minimum and maximum nonzero weights, respectively. If [Formula: see text] then all nonzero codewords of [Formula: see text] are minimal. In this paper, a necessary and sufficient condition is given for self-dual codes and for MDS codes to verify the inequality (*). Special codes are examined and applied for secret sharing schemes.

Sign in / Sign up

Export Citation Format

Share Document