scholarly journals Nonexistence of Certain Singly Even Self-Dual Codes with Minimal Shadow

10.37236/7155 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Stefka Bouyuklieva ◽  
Masaaki Harada ◽  
Akihiro Munemasa
Keyword(s):  
Minimum Weight ◽  
Dual Code ◽  
Weight Enumerator ◽  
Dual Codes

It is known that there is no extremal singly even self-dual $[n,n/2,d]$ code with minimal shadow for $(n,d)=(24m+2,4m+4)$, $(24m+4,4m+4)$, $(24m+6,4m+4)$, $(24m+10,4m+4)$ and $(24m+22,4m+6)$. In this paper, we study singly even self-dual codes with minimal shadow having minimum weight $d-2$ for these $(n,d)$. For $n=24m+2$, $24m+4$ and $24m+10$, we show that the weight enumerator of a singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow is uniquely determined and we also show that there is no singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow for $m \ge 155$, $m \ge 156$ and $m \ge 160$, respectively. We demonstrate that the weight enumerator of a singly even self-dual code with minimal shadow is not uniquely determined for parameters $[24m+6,12m+3,4m+2]$ and $[24m+22,12m+11,4m+4]$.

Extended maximal self-orthogonal codes

2019 ◽  
Vol 11 (05) ◽  
pp. 1950052
Author(s):  
Yilmaz Durğun
Keyword(s):  
Dual Code ◽  
Weight Enumerator ◽  
Hamming Weight ◽  
Golay Code ◽  
Orthogonal Codes ◽  
Orthogonal Code ◽  
Binary Golay Code

Self-dual and maximal self-orthogonal codes over [Formula: see text], where [Formula: see text] is even or [Formula: see text](mod 4), are extensively studied in this paper. We prove that every maximal self-orthogonal code can be extended to a self-dual code as in the case of binary Golay code. Using these results, we show that a self-dual code can also be constructed by gluing theory even if the sum of the lengths of the gluing components is odd. Furthermore, the (Hamming) weight enumerator [Formula: see text] of a self-dual code [Formula: see text] is given in terms of a maximal self-orthogonal code [Formula: see text], where [Formula: see text] is obtained by the extension of [Formula: see text].

A SPECIAL CLASS OF QUASI-CYCLIC CODES

2017 ◽  
Vol 96 (3) ◽  
pp. 513-518 ◽  
Author(s):  
MINJIA SHI ◽  
JIE TANG ◽  
MAORONG GE ◽  
LIN SOK ◽  
PATRICK SOLÉ
Keyword(s):  
Special Class ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Dual Code ◽  
Extension Field ◽  
Dual Codes ◽  
Lcd Codes ◽  
Quasi Cyclic Codes

We study a special class of quasi-cyclic codes, obtained from a cyclic code over an extension field of the alphabet field by taking its image on a basis. When the basis is equal to its dual, the dual code admits the same construction. We give some examples of self-dual codes and LCD codes obtained in this way.

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 A Characterization of Graphs by Codes from their Incidence Matrices

10.37236/2770 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Peter Dankelmann ◽  
Jennifer D. Key ◽  
Bernardo G. Rodrigues
Keyword(s):  
Incidence Matrix ◽  
Linear Codes ◽  
Minimum Weight ◽  
Weight Enumerator ◽  
Edge Connectivity ◽  
Connected Graphs ◽  
Incidence Matrices ◽  
Wide Range ◽  
The Matrix

We continue our earlier investigation of properties of linear codes generated by the rows of incidence matrices of $k$-regular connected graphs on $n$ vertices. The notion of edge connectivity is used to show that, for a wide range of such graphs, the $p$-ary code, for all primes $p$, from an $n \times \frac{1}{2}nk$ incidence matrix has dimension $n$ or $n-1$, minimum weight $k$, the minimum words are the scalar multiples of the rows, there is a gap in the weight enumerator between $k$ and $2k-2$, and the words of weight $2k-2$ are the scalar multiples of the differences of intersecting rows of the matrix. For such graphs, the graph can thus be retrieved from the code.

scholarly journals An upper bound for the minimum weight of the dual codes of desarguesian planes

2009 ◽  
Vol 30 (1) ◽  
pp. 220-229 ◽  
Author(s):  
J.D. Key ◽  
T.P. McDonough ◽  
V.C. Mavron
Keyword(s):  
Upper Bound ◽  
Minimum Weight ◽  
Dual Codes ◽  
Desarguesian Planes

Group Actions and Codes

2001 ◽  
Vol 53 (1) ◽  
pp. 212-224 ◽  
Author(s):  
V. Puppe
Keyword(s):  
Fixed Points ◽  
Equivariant Cohomology ◽  
Group Actions ◽  
Closed Manifold ◽  
Dual Code ◽  
Cohomology Algebra ◽  
Dual Codes ◽  
3 Dimensional

AbstractA 2-action with “maximal number of isolated fixed points” (i.e., with only isolated fixed points such that dimk(⊕iHi (M; k)) = |M2|, k = ) on a 3-dimensional, closed manifold determines a binary self-dual code of length = . In turn this code determines the cohomology algebra H*(M; k) and the equivariant cohomology . Hence, from results on binary self-dual codes one gets information about the cohomology type of 3-manifolds which admit involutions with maximal number of isolated fixed points. In particular, “most” cohomology types of closed 3-manifolds do not admit such involutions. Generalizations of the above result are possible in several directions, e.g., one gets that “most” cohomology types (over ) of closed 3-manifolds do not admit a non-trivial involution.

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