Linear Codes from Incidence Matrices of Unit Graphs
LLinear Codes from Incidence Matrices of Unit Graphs
N. AnnamalaiAssistant ProfessorIndian Institute of Information Technology KottayamPala, Kerala, IndiaEmail: [email protected] DurairajanAssociate ProfessorDepartment of MathematicsSchool of Mathematical SciencesBharathidasan UniversityTiruchirappalli-620024, Tamil Nadu, IndiaEmail: cdurai66@rediffmail.com
Running head:
Linear Codes from Incidence Matrices of Unit Graphs a r X i v : . [ c s . I T ] N ov bstract In this paper, we examine the binary linear codes with respect to Hammingmetric from incidence matrix of a unit graph G ( Z n ) with vertex set is Z n and twodistinct vertices x and y being adjacent if and only if x + y is unit. The mainparameters of the codes are given. Keywords:
Linear codes, Incidence matrix, Unit graph.
Let n be a positive integer and let Z n be the ring of integers modulo n. In [5], Ralph P.Grimaldi defined a graph G ( Z n ) based on the elements and units of Z n . The vertices ofthe unit graph G ( Z n ) are the elements of Z n and distinct vertices x and y are definedto be adjacent if and only if x + y is a unit of Z n . That is, xy is an edge if and only if x + y is a unit in Z n . For a positive integer m, it follows that G ( Z m ) is a φ (2 m )-regulargraph where φ is the Euler phi function.Codes generated by the incidence matrices and the adjacency matrices of variousgraphs are discussed in [4, 7, 8, 1].In what follows, all rings are associative with nonzero identity, denoted by 1, whichis preserved by ring homomorphisms and inherited by subrings. Also throughout thearticle, by a graph G we mean a finite undirected graph without loops or multiple edges(unless otherwise specified). In this section, we study the basic properties of unit graphs and linear codes.Let F q denote the finite field with cardinality q. Then F nq is a n -dimensional vectorspace over the finite field F q . The Hamming weight w H ( a ) of a ∈ F q is w H ( a ) = a = 01 for a (cid:54) = 0 . a, b ∈ F q , then the Hamming distance of a and b is defined to be d H ( a, b ) = w H ( a − b ) . For a word x = ( x , . . . , x n ) ∈ F nq , the Hamming weight w H ( x ) is defined to be w H ( x ) = n (cid:88) i =1 w H ( x i ) . Let x = ( x , . . . , x n ) , y = ( y , . . . , y n ) ∈ F nq , then the Hamming distance between x and y is defined by d H ( x, y ) = n (cid:88) i =1 d H ( x i , y i ) = n (cid:88) i =1 w H ( x i − y i ) . A non-empty subset C of F nq is said to be a q -ary code of length n. An element ofthe code C is called a codeword. A q -ary linear code C of length n is a subspace of thevector space F nq over F q . The minimum Hamming distance of a code C is defined by d H ( C ) = min c ,c ∈ C { d H ( c , c ) | c (cid:54) = c } . The minimum weight of a code C is the smallest among all weights of the non-zerocodewords of C. For q -ary linear code, we have d H ( C ) = w H ( C ) . All the codes here are linear codes and the notation [ n, k, d ] q will be used for a q -arycode of length n, dimension k and minimum distance d. A linear [ n, k, d ] q code is said tobe a Maximum Distance Separable(MDS) code if d = n − k + 1 . A generator matrix B for a linear code C is a matrix whose rows form a basis for the subspace C and C q ( B )is a code generated by matrix B over a finite field F q . Let G = ( V, E ) be a graph with vertex set V, edge set E and for any x, y ∈ V, [ x, y ]is denoted by an edge between x and y. An incidence matrix of G is a | V | × | E | matrix H with rows labelled by the vertices and columns by the edges and entries h ij = 1 if i thvertex is adjacent to the j th edge and h ij = 0 otherwise.Let R be a ring and let U ( R ) be the set of unit elements of R. The unit graph of R, denoted G ( R ) , is the graph obtained by setting all the elements of R to be the verticesand defining distinct vertices x and y to be adjacent if and only if x + y ∈ U ( R ) . Let G be a graph. We denote V ( G ) as the vetex set of G and E ( G ) as the edge setof G. Let x ∈ V ( G ) , then the degree of x, denoted deg ( x ) , is the number of edges of G incident with x. A graph G is said to be a r -regular if the degree of each vertex of G is equal to r. Let
W, X ⊆ V with W ∩ X = ∅ and let E ( W, X ) be the set of edges that have oneend in W and the other end in X. Write | E ( W, X ) | = q ( W, X ) . An edge-cut of a connected graph G is the set S ⊆ E such that G − S = ( V, E − S )is disconnected. 3he edge-connectivity λ ( G ) is the minimum cardinality of an edge-cut. That is, λ ( G ) = min ∅(cid:54) = W (cid:40) V q ( W, V − W ) . For any connected graph G, we have λ ( G ) ≤ δ ( G ) where δ ( G ) is minimum degree of thegraph G. Theorem 2.1. [2] Let R be a finite ring. Then the following statements hold for theunit graph of R. (a) If / ∈ U ( R ) , then the unit graph G ( R ) is a | U ( R ) | -regular graph.(b) If ∈ U ( R ) , then for every x ∈ U ( R ) we have deg ( x ) = | U ( R ) | − and for every x ∈ R \ U ( R ) we have deg ( x ) = | U ( R ) | . Theorem 2.2. [4] Let G = ( V, E ) be a connected graph and let H be a | V |×| E | incidencematrix for G. Then1. the code C ( H ) is a [ | E | , | V | − , λ ( G )] code2. the code C p ( H ) is [ | E | , | V | − , λ ( G )] p code for an odd prime p and G is bipartite. In this paper, we obtain a linear codes from the incidence matrix of the unit graph G ( Z n ) over F and we determined the main parameters of the code. In section 3, wediscussed codes from incidence matrix of a unit graph G ( Z p ) over the field F . In section4, we discussed codes from incidence matrix of a unit graph G ( Z p ) over the finite field F . All the unit graphs considered in this article is a simple and undirected graph. G ( Z p ) In this section, we study a binary linear code obtained from the incidence matrix of theunit graph G ( Z p ) where p is an odd prime and find its parameters.The units of Z p is U ( Z p ) = { , , · · · , p − } . Let G ( Z p ) be the unit graph withvertex set V = Z p and two distinct vertices x and y are adjacent if and only if x + y ∈ U ( Z p ) . Theorem 3.1.
Let G ( Z p ) = ( V, E ) be a unit graph. Then the graph is connected with | V | = p and | E | = ( p − . roof. Let G ( Z p ) be a unit graph. Then by definition, V = Z p . Since all non-zeroelements are units, 0 and y are adjacent for all nonzero y ∈ Z p . Therefore the graph isa connected graph. (cid:12)(cid:12)(cid:12)(cid:110) [0 , y ] | y ∈ Z p \ { } (cid:111)(cid:12)(cid:12)(cid:12) = p − (cid:12)(cid:12)(cid:12)(cid:8) [ j, y ] | y ∈ Z p \ { , , · · · , j, p − j } (cid:9)(cid:12)(cid:12)(cid:12) = p − ( j + 2) for 1 ≤ j ≤ p − p + 12 ≤ j ≤ p − , we have (cid:12)(cid:12)(cid:12)(cid:8) [ j, y ] | j < y (cid:9)(cid:12)(cid:12)(cid:12) = p − ( j + 1) . Therefore, | E | = p − p − (cid:88) j =1 p − ( j + 2) + p − (cid:88) j =1 j = ( p − . Hence, the unit graph G ( Z p ) is a connected graph with p vertices and ( p − Theorem 3.2.
Let G ( Z p ) be the unit graph. Then the edge-connectivity λ ( G ( Z p )) ofthe unit graph G ( Z p ) is p − . Proof. If W = { } ⊂ V, then q ( W, V − W ) = p − . If W = { a } ⊂ V, a (cid:54) = 0 , then q ( W, V − W ) = p − . If W ⊂ V with | W | > , then q ( W, V − W ) > p − . Therefore, λ ( G ( Z p )) = p − . As a consequence of above theorems, we have
Theorem 3.3.
Let p be an odd prime. Then the code generated by the incidence matrix H of the unit graph G ( Z p ) is a C ( H ) = (cid:104) ( p − , p − , p − (cid:105) code over the finitefield F . Proof.
Let G ( Z p ) be a unit graph and let H be the incidence matrix of G ( Z p ) . Since G ( Z p ) is a connected graph, by Theorem 2.2, the code C ( H ) is a [ | E | , | V |− , λ ( G ( Z p ))] code. By Theorem 3.1, we have | E | = ( p − | V | = p . By Theorem 4.1, theedge-connective of the unit graph G ( Z p ) is p − C ( H ) is a (cid:104) ( p − , p − , p − (cid:105) code. Example 3.4.
The unit graph G ( Z ) = ( V, E ) with | V | = 5 and | E | = (5 − = 8 is bb b bb hen the incidence matrix of the unit graph G ( Z ) is H = × Any four rows of H are linearly independent and the graph is connected, then the di-mension of the code C ( H ) over the finite field F generated by H is . The minimumdistance of the code C ( H ) is . Hence C ( H ) is an [ n, k, d ] = [8 , , code. G ( Z p ) In this section, we study the linear codes obtained from the incidence matrix of the unitgraph G ( Z p ) where p is an odd prime over F and we find the parameters of the code.The units of Z p is U ( Z p ) = { k ∈ Z p | ( k, p ) = 1 } and | U ( Z p ) | = p − . Theorem 4.1.
Let G ( Z p ) = ( V, E ) be a unit graph. Then | V | = 2 p and | E | = p ( p − . Proof.
Let G ( Z p ) be a unit graph. Then by definition, V = Z p . Since (2 , p ) = 2 (cid:54) = 1 , by Theorem 2.1, G ( Z p ) is a | U ( Z p ) | = ( p − p − . Since the number of edges of the k -regular graph with n vertices is nk , implies | E | = p ( p − . Theorem 4.2.
Let p be an odd prime. Then the code generated by the incidence matrix H of the unit graph G ( Z p ) is a C ( H ) = (cid:104) p ( p − , p − , p − (cid:105) code over thefinite field F . Proof.
Let G ( Z p ) be a unit graph and let H be the incidence matrix of G ( Z p ) . Since G ( Z p ) is a connected graph, by Theorem 2.2, the code C ( H ) is a [ | E | , | V | − , λ ( G ( Z p ))] code. Since G ( Z p ) is ( p − λ ( G ( Z p )) of the unit graph G ( Z p ) is p − H is p − . Since | E | = p ( p −
1) and | V | = 2 p , the main parametersof the code C ( H ) is [ p ( p − , p − , p − . Example 4.3.
The unit graph G ( Z ) = ( V, E ) with | V | = 6 and | E | = 6 is bbb bb Then the incidence matrix of the unit graph G ( Z ) is H = × Any five rows of H are linearly independent and the graph is connected, then the di-mension of the code C ( H ) over the finite field F generated by H is . The minimumdistance of the code C ( H ) is . Hence C ( H ) is an [ n, k, d ] = [6 , , code. Since d = n − k + 1 , the code C ( H ) is a MDS code. Conclusion
In this paper, we studied the codes generated by the incidence matrix of the unit graphof the different commutative rings with unity. Also we found the main parameters ofthe code over finite field F . We have consider only simple and undirected graphs in thisarticle. Finding the covering radius of these codes is the further direction to work.
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