scholarly journals Near-Extremal Type I Self-Dual Codes with Minimal Shadow over GF(2) and GF(4)

Information ◽  
2018 ◽  
Vol 9 (7) ◽  
pp. 172
Author(s):  
Sunghyu Han
Keyword(s):  
Dual Code ◽  
Type I ◽  
Type Ii ◽  
Dual Codes ◽  
Comprehensive Description ◽  
Type Ii Codes

Binary self-dual codes and additive self-dual codes over GF(4) contain common points. Both have Type I codes and Type II codes, as well as shadow codes. In this paper, we provide a comprehensive description of extremal and near-extremal Type I codes over GF(2) and GF(4) with minimal shadow. In particular, we prove that there is no near-extremal Type I [24m,12m,2m+2] binary self-dual code with minimal shadow if m≥323, and we prove that there is no near-extremal Type I (6m+1,26m+1,2m+1) additive self-dual code over GF(4) with minimal shadow if m≥22.

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 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.

Formally Self-Dual Codes Related to Type II Codes

2003 ◽  
Vol 14 (2) ◽  
pp. 81-88 ◽  
Author(s):  
Koichi Betsumiya ◽  
Masaaki Harada
Keyword(s):  
Type Ii ◽  
Dual Codes ◽  
Type Ii Codes

EISENSTEIN LATTICES, GALOIS RINGS AND QUATERNARY CODES

2006 ◽  
Vol 02 (02) ◽  
pp. 289-303 ◽  
Author(s):  
PHILIPPE GABORIT ◽  
ANN MARIE NATIVIDAD ◽  
PATRICK SOLÉ
Keyword(s):  
Modular Lattice ◽  
Orthogonal Basis ◽  
Galois Ring ◽  
Galois Rings ◽  
Type Ii ◽  
Dual Codes ◽  
Type Ii Codes ◽  
Quaternary Codes ◽  
Double Circulant Codes ◽  
Circulant Codes

Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) ℤ4-codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual ℤ4-codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual 𝔽4-codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.

Self-dual codes over ℤ4 + wℤ4

2015 ◽  
Vol 07 (02) ◽  
pp. 1550014 ◽  
Author(s):  
Rama Krishna Bandi ◽  
Maheshanand Bhaintwal
Keyword(s):  
Linear Codes ◽  
Construction Method ◽  
Type Ii ◽  
Dual Codes ◽  
Type Ii Codes

In this paper we study linear codes over the rings Ri = ℤ4 + wℤ4, where w2 = i, i = 1, 2w. We characterize self-dual codes over Ri and propose a construction method for self-dual codes over these rings. We also briefly discuss circulant self-dual codes and Type II codes. Some examples are given.

Byte weight enumerators and modular forms of genus r

2015 ◽  
Vol 14 (06) ◽  
pp. 1550080
Author(s):  
Anuradha Sharma ◽  
Amit K. Sharma
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Zeta Functions ◽  
Siegel Modular Forms ◽  
Type I ◽  
Type Ii ◽  
Jacobi Forms ◽  
Weight Enumerators ◽  
Type Ii Codes ◽  
Real Quaternions

For a positive integer m, let R be either the ring ℤ2m of integers modulo 2m or the quaternionic ring Σ2m = ℤ2m + αℤ2m + βℤ2m + γℤ2m with α = 1 + î, β = 1 + ĵ and [Formula: see text], where [Formula: see text] are elements of the ring ℍ of real quaternions satisfying [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we obtain Jacobi forms (or Siegel modular forms) of genus r from byte weight enumerators (or symmetrized byte weight enumerators) in genus r of Type I and Type II codes over R. Furthermore, we derive a functional equation for partial Epstein zeta functions, which are summands of classical Epstein zeta functions associated with quadratic forms.

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.

Heat-Etching with a Bullivant Type II Simple Freeze-Cleave Device

1970 ◽  
Vol 28 ◽  
pp. 106-107 ◽  
Author(s):  
Ronald S. Weinstein ◽  
N. Scott McNutt
Keyword(s):  
New Zealand ◽  
General Hospital ◽  
Massachusetts General Hospital ◽  
Type I ◽  
Type Ii ◽  
Collaborative Effort ◽  
Temperature And Pressure ◽  
The University

The Type I simple cold block device was described by Bullivant and Ames in 1966 and represented the product of the first successful effort to simplify the equipment required to do sophisticated freeze-cleave techniques. Bullivant, Weinstein and Someda described the Type II device which is a modification of the Type I device and was developed as a collaborative effort at the Massachusetts General Hospital and the University of Auckland, New Zealand. The modifications reduced specimen contamination and provided controlled specimen warming for heat-etching of fracture faces. We have now tested the Mass. General Hospital version of the Type II device (called the “Type II-MGH device”) on a wide variety of biological specimens and have established temperature and pressure curves for routine heat-etching with the device.

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