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.

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

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.

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.

scholarly journals Mass Formula for Self-Dual Codes over Galois Rings GR(p3, r)

2019 ◽  
Vol 12 (4) ◽  
pp. 1701-1716
Author(s):  
Trilbe Lizann Espina Vasquez ◽  
Gaudencio Jr. Cempron Petalcorin
Keyword(s):  
Positive Integer ◽  
Mass Formula ◽  
Galois Ring ◽  
The Self ◽  
Galois Rings ◽  
Dual Codes ◽  
Characteristic P

Let $p$ be an odd prime and $r$ a positive integer. Let $\text{GR}(p^3,r)$ be the Galois ring of characteristic $p^3$ and cardinality $p^{3r}$. In this paper, we investigate the self-dual codes over $\text{GR}(p^3,r)$ and give a method to construct self-dual codes over this ring. We establish a mass formula for self-dual codes over $\text{GR}(p^3,r)$ and classify self-dual codes over $\text{GR}(p^3,2)$ of length 4 for $p=3,5$.

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 Normal Bases on Galois Ring Extensions

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 702
Author(s):  
Aixian Zhang ◽  
Keqin Feng
Keyword(s):  
Signal Processing ◽  
Finite Field ◽  
Block Cipher ◽  
Field Extension ◽  
Galois Ring ◽  
Normal Basis ◽  
Galois Rings ◽  
Ring Extension ◽  
Galois Fields ◽  
Normal Bases

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.

scholarly journals Construction of MDS self-dual codes over Galois rings

2007 ◽  
Vol 45 (2) ◽  
pp. 247-258 ◽  
Author(s):  
Jon-Lark Kim ◽  
Yoonjin Lee
Keyword(s):  
Galois Rings ◽  
Dual Codes

Symbol-Pair Distances of Repeated-Root Negacyclic Codes of Length 2s over Galois Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 581-600
Author(s):  
Hai Q. Dinh ◽  
Hualu Liu ◽  
Roengchai Tansuchat ◽  
Thang M. Vo
Keyword(s):  
Finite Field ◽  
Galois Ring ◽  
Maximum Distance ◽  
Galois Rings ◽  
Constacyclic Codes ◽  
Chain Ring ◽  
Negacyclic Codes ◽  
Singleton Bound ◽  
Maximum Distance Separable ◽  
Special Case

Negacyclic codes of length [Formula: see text] over the Galois ring [Formula: see text] are linearly ordered under set-theoretic inclusion, i.e., they are the ideals [Formula: see text], [Formula: see text], of the chain ring [Formula: see text]. This structure is used to obtain the symbol-pair distances of all such negacyclic codes. Among others, for the special case when the alphabet is the finite field [Formula: see text] (i.e., [Formula: see text]), the symbol-pair distance distribution of constacyclic codes over [Formula: see text] verifies the Singleton bound for such symbol-pair codes, and provides all maximum distance separable symbol-pair constacyclic codes of length [Formula: see text] over [Formula: see text].

Type II Codes over IF2r

2001 ◽  
pp. 102-111 ◽  
Author(s):  
Koichi Betsumiya ◽  
Masaaki Harada ◽  
Akihiro Munemasa
Keyword(s):  
Type Ii ◽  
Type Ii Codes
Sign in / Sign up

Export Citation Format

Share Document