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Perfect codes over graphs

Jan Kratochvíl
1986 Journal of combinatorial theory. Series B (Print)  
We present a generalization of the notion of perfect codes: perfect codes over graphs.  ...  We show an infinite family of l-perfect codes in second powers of graphs and we prove the nonexistence of nontrivial l-perfect codes over complete bipartite graphs with at least three vertices.  ...  NeSeti-il: perfect codes over graphs, which correspond to perfect codes over structured alphabets. The classical perfect codes are then just perfect codes over complete graphs.  ... 
doi:10.1016/0095-8956(86)90079-1 fatcat:bt6eqq5t7vdjhazn3fwluopb5e

Perfect Codes Over Induced Subgraphs of Unit Graphs of Ring of Integers Modulo n

Mohammad Hassan Mudaber, Nor Haniza Sarmin, Ibrahim Gambo
2021 WSEAS Transactions on Mathematics  
The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes  ...  In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined.  ...  The results that have been ob tained represent the types of the induced subgraphs of the unit graphs as well as the perfect codes over these graphs.  ... 
doi:10.37394/23206.2021.20.41 fatcat:eliot2vfa5cxrlmox64lgwabfy

Quotients of Gaussian graphs and their application to perfect codes

C. Martínez, R. Beivide, C. Camarero, E. Stafford, E.M. Gabidulin
2010 Journal of symbolic computation  
A graph-based model of perfect two-dimensional codes is presented in this work. This model facilitates the study of the metric properties of the codes.  ...  Signal spaces are modeled by means of Cayley graphs defined over the Gaussian integers and denoted as Gaussian graphs.  ...  This graph theory approach determined the existence and construction of perfect codes over the Gaussian integers.  ... 
doi:10.1016/j.jsc.2010.03.013 fatcat:wulopenl4fftpfcngbyh2a2f3a

Perfect codes in Doob graphs

Denis S. Krotov
2015 Designs, Codes and Cryptography  
Keywords: perfect codes, Doob graphs, distance regular graphs.  ...  We study 1-perfect codes in Doob graphs D(m,n). We show that such codes that are linear over GR(4^2) exist if and only if n=(4^g+d-1)/3 and m=(4^g+2d-4^g+d)/6 for some integers g > 0 and d>0.  ...  Additionally, we study the existence of linear, over the rings GR(4 2 ) and Z 4 , 1-perfect codes in Doob graphs.  ... 
doi:10.1007/s10623-015-0066-6 fatcat:y5glds26dnclbetipy6gwivhsa

On weight distributions of perfect colorings and completely regular codes

Denis S. Krotov
2010 Designs, Codes and Cryptography  
Codewords: completely regular code; equitable partition; partition design; perfect coloring; perfect structure; regular partition; weight distribution; weight enumerator.  ...  We show how to compute this distribution by the knowledge of the color composition over the set.  ...  So, given a completely regular code C, we also have a way to reconstruct the weight distribution with respect to C of any other perfect structure (perfect coloring) f over the same graph by knowledge of  ... 
doi:10.1007/s10623-010-9479-4 fatcat:45tzcrplifhdxecwpdldmzrrii

Quasi-Perfect Lee Codes of Radius 2 and Arbitrarily Large Dimension

Cristobal Camarero, Carmen Martinez
2016 IEEE Transactions on Information Theory  
A construction of 2-quasi-perfect Lee codes is given over the space Z_p^n for p prime, p≡± 512 and n=2[p/4]. It is known that there are infinitely many such primes.  ...  A series of computations show that related graphs are Ramanujan, which could provide further connections between Coding and Graph Theories.  ...  Huber in [20] gave 1-perfect codes over Gaussian integers and some non-perfect codes with greater correction.  ... 
doi:10.1109/tit.2016.2517069 fatcat:ogbmi3cb35cb5ctc4avf5vzgvu

The computational strength of matchings in countable graphs [article]

Stephen Flood, Matthew Jura, Oscar Levin, Tyler Markkanen
2020 arXiv   pre-print
In a 1977 paper, Steffens identified an elegant criterion for determining when a countable graph has a perfect matching.  ...  The results of this paper explore the relationship between graph theory and logic by showing the way in which specific changes to a single graph-theoretic principle impact the corresponding proof-theoretical  ...  If a countable graph G has a perfect matching, then G satisfies condition (A). Furthermore, this holds over RCA 0 . Proof. Fix a graph G and a perfect matching N .  ... 
arXiv:2006.11334v1 fatcat:gnjsgfyngnajbkswztsoua2wdy

Additive perfect codes in Doob graphs [article]

Minjia Shi Anhui University, Hefei, China, Sobolev Institute of Mathematics, Novosibirsk, Russia)
2018 arXiv   pre-print
We construct a 3-parameter class of additive perfect codes in Doob graphs and show that the known necessary conditions of the existence of additive 1-perfect codes in D(m,n'+n") are sufficient.  ...  Additionally, two quasi-cyclic additive 1-perfect codes are constructed in D(155,0+31) and D(2667,0+127).  ...  In [6] , the author completely solved the problem of existence of linear 1-perfect codes in Doob graphs (a linear code in Doob graph forms a module over the Galois ring GR(4 2 )) and proposed an open  ... 
arXiv:1806.04834v1 fatcat:7x4irnpribfzpg45epvezvjkde

Quasi-Perfect Lee Codes from Quadratic Curves over Finite Fields [article]

Sihem Mesnager, Chunming Tang, Yanfeng Qi
2016 arXiv   pre-print
Our approach uses subsets derived from some quadratic curves over finite fields (in odd characteristic) to derive two classes of 2-quasi-perfect Lee codes are given over the space Z_p^n for n=p^k+1/2 (  ...  In this paper we firstly highlight the relationships between subset sums, Cayley graphs, and Lee linear codes and present some results.  ...  [5] , where quasi-perfect codes over Z and QP L(n, t) codes have been introduced). Also, in [14] the authors presented some quasi-perfect codes for n = 3 and few radii.  ... 
arXiv:1608.06748v1 fatcat:dplbtuzhrnhyfn5ovknlved7aa

Perfect codes in vertex-transitive graphs [article]

Yuting Wang, Junyang Zhang
2021 arXiv   pre-print
Given a graph Γ, a perfect code in Γ is an independent set C of vertices of Γ such that every vertex outside of C is adjacent to a unique vertex in C, and a total perfect code in Γ is a set C of vertices  ...  To study (total) perfect codes in vertex-transitive graphs, we generalize the concept of subgroup (total) perfect code of a finite group introduced in as follows: Given a finite group G and a subgroup  ...  Similarly, the perfect t-error-correcting Lee codes over Z m (m ≥ 3) are precisely the perfect t-codes in L(n, m).  ... 
arXiv:2112.06236v1 fatcat:ygj7qld5ezeydb3jd63w4ogwje

On perfect codes in Cartesian products of graphs

Michel Mollard
2011 European journal of combinatorics (Print)  
Assuming the existence of a partition in perfect codes of the vertex set of a finite or infinite bipartite graph G we give the construction of a perfect code in the Cartesian product G G P 2 .  ...  Such a partition is easily obtained in the case of perfect codes in Abelian Cayley graphs and we give some examples of applications of this result and its generalizations.  ...  It is clear that, conversely, from a partition of the vertex set of a graph in perfect codes we obtain a code-coloring of this graph.  ... 
doi:10.1016/j.ejc.2010.11.007 fatcat:anltxzft7nepbfteifospozkve

The existence of perfect codes in Doob graphs [article]

Denis S. Krotov (Sobolev Institute of Mathematics, Novosibirsk, Russia)
2018 arXiv   pre-print
We solve the problem of existence of perfect codes in the Doob graph.  ...  It is shown that 1-perfect codes in the Doob graph D(m,n) exist if and only if 6m+3n+1 is a power of 2; that is, if the size of a 1-ball divides the number of vertices.  ...  So, if there is a 1-perfect code in a 4-ary Hamming graph, then there is a 1-perfect code in every Doob graph of the same diameter. Proof.  ... 
arXiv:1810.03772v1 fatcat:43rvu7xbxnd3bm533xmqlmvwse

Perfect One-Factorization Conjecture

Chriestie Montolalu
2015 d'CARTESIAN  
Perfect one-factorization of the complete graph K2n for all n greater and equal to 2 is conjectured. Nevertheless some families of complete graphs were found to have perfect one-factorization.  ...  This paper will show some of the perfect one-factorization results in some families of complete graph as well as some result in application. Keywords: complete graph, one-factorization  ...  Let c(F ) to be the number of perfect pairs of F , and c(G) to be the maximum c(F ) over all one-factorizations F of G.  ... 
doi:10.35799/dc.4.1.2015.8340 fatcat:cumqcytby5aitgs5ucdewh347u

Resource placement in torus-based networks

M.M. Bae
1997 IEEE transactions on computers  
This paper contains some solutions to perfect distance-t and perfect / quasi-perfect j-adjacency placement in a k-ary n-cube and a torus using Lee distance error-correcting codes.  ...  A vector c of length n over Z n k is a code word More on these codes will be described in Section 3.1. iff cH T = 0.  ...  For example, the set of words C = f00; 12; 24; 32; 41g over Z 2 5 form a Lee distance 3 code; the H matrix for this code is H = [ 3 1 ].  ... 
doi:10.1109/12.628393 fatcat:hxszykwnxvbxzeg46hlj6376q4

On Index coding for Complementary Graphs with focus on Circular Perfect Graphs [article]

Bhavana M, Prasad Krishnan
2019 arXiv   pre-print
They form a strict superclass of the perfect graphs, whose index coding broadcast rates are well known.  ...  We present the broadcast rate of index coding for side-information graphs whose complements are circular perfect, along with an optimal achievable scheme.  ...  We construct a linear code of rate k d over any field F q . Suppose that each message is a vector of length d.  ... 
arXiv:1901.05898v1 fatcat:de3vjaqu4jhexp7uitde7kxvne
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