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 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.

Type I and Type II codes over the ring 𝔽2 + u𝔽2 + v𝔽2 + uv𝔽2

2019 ◽  
Vol 12 (02) ◽  
pp. 1950025 ◽  
Author(s):  
Ankur ◽  
Pramod Kumar Kewat
Keyword(s):  
Type I ◽  
Type Ii ◽  
Dual Codes ◽  
Generator Matrix ◽  
Type Ii Codes

We discuss self-dual codes over the ring [Formula: see text]. We characterize the structure of self-dual, Type I codes and Type II codes over [Formula: see text] with given generator matrix in terms of the structure of their Torsion and Residue codes. We construct self-dual, Type I and Type II codes over [Formula: see text] for different lengths.

scholarly journals Group matrix ring codes and constructions of self-dual codes

2021 ◽  
Author(s):  
S. T. Dougherty ◽  
Adrian Korban ◽  
Serap Şahinkaya ◽  
Deniz Ustun
Keyword(s):  
Matrix Ring ◽  
Type I ◽  
Type Ii ◽  
Dual Codes ◽  
Generator Matrix ◽  
Matrix Rings ◽  
Group Matrix ◽  
The Matrix ◽  
New Type ◽  
Matrix Construction

AbstractIn this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring $$M_k(R)$$ M k ( R ) and the ring R,  where R is the commutative Frobenius ring. We show that codes over the ring $$M_k(R)$$ M k ( R ) are one sided ideals in the group matrix ring $$M_k(R)G$$ M k ( R ) G and the corresponding codes over the ring R are $$G^k$$ G k -codes of length kn. Additionally, we give a generator matrix for self-dual codes, which consist of the mentioned above matrix construction. We employ this generator matrix to search for binary self-dual codes with parameters [72, 36, 12] and find new singly-even and doubly-even codes of this type. In particular, we construct 16 new Type I and 4 new Type II binary [72, 36, 12] self-dual codes.

scholarly journals Results on Binary Linear Codes With Minimum Distance 8 and 10

2011 ◽  
Vol 57 (9) ◽  
pp. 6089-6093 ◽  
Author(s):  
Iliya Georgiev Bouyukliev ◽  
Erik Jacobsson
Keyword(s):  
Minimum Distance ◽  
Linear Codes ◽  
Binary Linear 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].

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