Graph Representation of Binary Greedy Codes, Coloring and Decoding

European Journal of Scientific Research ISSN 1450-216X Vol.30 No.4 (2009), pp.579-583 © EuroJournals Publishing, Inc. 2009 http://www.eurojournals.com/ejsr.htm Graph Representation of Binary Greedy Codes, Coloring and Decoding Shoaib U Din Department of Mathematics, University of the Punjab Lahore E-mail: shoaibdin@gmail.com Khalil Ahmad Department of Mathematics, University of Management and Technology Lahore E-mail: khalil@umt.edu.pk Abstract Error-correcting codes are widely used to increase the reliability of transmission of information over various forms of communication channels. Codes with a given minimum distance d and length n can be constructed by a greedy algorithm [2]. Let binary vectors of length n represents vertices of a graph, vertex coloring of the graph partitions it into color classes. In this paper, I propose an algorithm for the allocation of colors to the binary vectors (vertices of graph). Zero color class infact represents greedy code. A code is constructed to demonstrate a color decoding technique for linear codes in addition to the schemes already known for decoding such as Syndrome Decoding Array (S.D.A) [9]. Keywords: B-ordering, vertex coloring , Linear codes Introduction There are a number of ways to generate a code, however in this paper greedy codes are discussed only. Greedy codes are the codes generated by the application of greedy algorithm on the vectors arranged in B-ordering [2]. A graph G (V, E) is constructed keeping in view the parameters of the code. Where binary vectors of length n represents vertices of G and any tow vertices are connected by an edge if distance between the vertices is less than or equal to d (minimum distance of the code). Each vertex is assigned a non-ive integer as its color using the greedy algorithm. The concepts of modern algebra are used to assign colors to the groups of vertices instead of assigning colors to the vertices one by one. The vertices with color zero represent the greedy code itself. A decoding algorithm based on vertex coloring is presented. Greedy codes A greedy code is a code generated by the application of a greedy algorithm, where the vectors are arranged in a certain type of ordering. When first vector in the list having a given property is selected, the algorithm continues recursively, until the list is exhausted. The problem is to maximize the number of vectors in the code. The use of greedy algorithm for generating codes is especially attractive due to its natural way of producing a code with a given minimum distance. The vectors are arranged in an ordering and each vector is analyzed in turn for being accepted or rejected according to its distance from vectors already chosen. Graph Representation of Binary Greedy Codes, Coloring and Decoding 580 The ordering is important and it happens that the B-ordering, a very natural type of ordering, gives a linear code every time [2] [5]. B-ordering in the binary case B-ordering is a generalization of lexicographic ordering[7]. It was first defined and discussed by Brualdi and Pless in [2]. We get a B-ordering of all binary n-tuples by choosing an ordered basis {b1,b2, ……bn} of vn. The first vector in the B-ordering is the zero vector, and the next is b1,b2, b2⊕b1 ,b3,b3⊕b1, b3⊕b2, b3⊕b2⊕b1, b4, ……… where if the first 2i-1 vectors of the ordering have been generated using B- basis elements b1,b2, ….bi-1, then the next 2i-1 vectors are generated by adding bi to those vectors already produced in order. Two different B-ordering of vn are obtained by choosing two different ordered bases[4]. Given a minimum distance d, choose a set of vectors C with the zero vector first; then go through the vectors in B-ordering and choose the next which has distance d or more from all vectors already chosen. The surprising result is that C is a linear code [2]. Codes found in this fashion are called greedy codes. Greedy codes are generated by applying greedy algorithm on vectors arranged in B-ordering. Codes constructed via this algorithm have been optimal or near-optimal, with dimensions either the highest possible for a given length and minimum distance or within one of the highest possible. In fact such codes satisfy Varshamove – Gilbert bound, which is the best known lower bound. W Chen[5],[6] used greedy algorithm to find the weight hierarchies of binary linear codes of dimension 4. Graph coloring A graph G(V n ,E)is constructed keeping in view the parameters of a code C(n,k,d). The vertex set V n 2 2 n (G) consists of binary vectors of length n, according to their B-ordering i.e. V 2 ={v0,v1,v2 …. v 2 n −1 }. E={eij |d(vi,vj) ≤ d-1} Where d(vi,vj)= the number of places in which vi and vj differ. A vertex coloring of a graph G =(V n , E) is a map c: V n → S s.t. c (v) ≠ c (w) when ever v 2 2 and w are adjacent. The elements of the set S are called the available colors. All that interest us about S is its size, typically, we shall be asking for the smallest integer k such that G has a k – coloring, a vertex coloring c: V n → {1,… k}. This k is the (vertex-) chromatic number of G; it is denoted by χ 2 (G). A graph G with χ (G) =k is called k – chromatic; if χ (G) ≤ k, we call G k – colorable. Note that a k – coloring is nothing but a vertex partition into k independent sets , now called color classes; the non – trivial 2 – colorable graphs , for example , are precisely the bipartite graphs. Now we assign to each vertex in V n a non negative integer, called a color. Colors may be 2 assigned to the vertices in the following recursive manner. Each vertex is considered in order and the first vertex, which is the zero vertex v0 , is assigned 0 as its color; then if v is a vertex under consideration and G is the set of colors of all previous vectors which are at distance less than d from v, then the color of v is the least nonnegative integer which is not an element of G. The set of all vertices having color 0, is infect the greedy code itself. Thus the assignment of colors is a generalization of the basic greedy algorithm for generating codes. The colors may also be assigned in order, as opposed to assigning to each vertex in order a color. One may first select all vertices with color 0 (the code); then select all vertices with color 1, etc. rather than assigning the first vertex a color, then the second vertex its color, etc. Either method produces the same assignment of colors 581 Shoaib U Din and Khalil Ahmad Let V n is the set of all binary strings of length n, it qualifies for a ring under the operations of 2 + 0 1 . 0 1 addition 0 0 1 and multiplication 0 0 0 1 1 0 1 0 1 It can be proved that c: V 2 → S is a homomorphism and its kernel is a binary greedy code. Instead of assigning colors to all vertices in V one by one we only need to evaluate kernel of c, then we capture all the pre-images of a mapping c by additively translating the kernel. Theorem m m Let c: V 2 → S be a homomorphism from vector space V 2 to vector space S and s ∈ c (V). Then −1 −1 c (s) equals the coset, kernel (c)+ v, where v is any given element of c (s). n n n Proof −1 Let s c (V), and choose any v ∈ c (s) (which is non-empty by assumption). We must show that the −1 sets ker(c) + v and c (s) are equal. Choose an arbitrary element a + v ∈ ker(c) + v, where a ∈ ker(c). Then c(a + v) = c(a) +c(v) = 0 + c(v) = s. −1 Thus, a + v ∈ c (s), as claimed. −1 Conversely, choose t ∈ c (s). Then consider t – v; c (t – v) = c(t) - c(v) = s – s = 0, and so t – v ∈ ker(c). But then t = (t – v) + v ∈ ker(c) + v, as required. By definition of function pre-images of distinct elements are disjoint from one another, so the set of cosets of kernel decomposes the domain space into set of pair wise disjoint subsets. In other words the set of cosets of kernel partitions the graph G onto color classes. The unique one of these sets containing ‘0’ is a subspace since a subspace has to contain zero it is clear that only one of the cosets can be a subspace. One can think of a subspace itself as a coset, ker (c)+0 comprise binary greedy code is a subspace of a vector space consists of all strings of length n. Hamming greedy codes We can produce perfect greedy codes with the parameters of Hamming codes via greedy algorithm. Let all binary n-tupples vn be in their B-ordering and choose a distance d. We construct a set C consisting of the vectors with g-value zero; C is actually a linear code (n, k, d). With d=3, n=2r-1 r>2 k=2r-1-r Hamming (2r-1, 2r-1 -r,3) codes are obtained. For example let r =3 then n = 7; k = 4; d = 3, (7,4,3) Hamming code is produced. Hamming greedy code (7,4,3) generated via Greedy Algorithm B = { 0000001, 0000011, 0000111, 0001111, 0011111, 0111111, 1111111} By definition of greedy codes C consists of all words with color 0. C = {0000000, 0000111, 0011110, 0011001, 0110011, 0110100, 0101101, 0101010, 1111111, 1111000, 1100001, 1100110, 1001100, 1001011, 1010010, 1010101} We can arrange above data w .r .t. colors. Graph Representation of Binary Greedy Codes, Coloring and Decoding Table 1: gvalue S. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 582 0 0000000 0000111 0011110 0011001 0110011 0110100 0101101 0101010 1111111 1111000 1100001 1100110 1001100 1001011 1010010 1010101 1 0000001 0000110 0011111 0011000 0110010 0110101 0101100 0101011 1111110 1111001 1100000 1100111 1001101 1001010 1010011 1010100 2 0000011 0000100 0011101 0011010 0110000 0110111 0101110 0101001 1111100 1111011 1100010 1100101 1001111 1001000 1010001 1010110 3 0000010 0000101 0011100 0011011 0110001 0110110 0101111 0101000 1111101 1111010 1100010 1100100 1001110 1001001 1010000 1010111 4 0001111 0001000 0010001 0010110 0111100 0111011 0100010 0100101 1110000 1110111 1101110 1101001 1000011 1000100 1011101 1011010 5 0001110 0001001 0010000 0010111 0111101 0111010 0100011 0100100 1110001 1110110 1101111 1101000 1000010 1000101 1011100 1011011 6 0001100 0001011 0010010 0010101 0111111 0111000 0100001 0100110 1110011 1110100 1101101 1101010 1000000 1000111 1011110 1011001 7 0001101 0001010 0010011 0010100 0111110 0111001 0100000 0100111 1110010 1110101 1101100 1101011 1000001 1000110 1011111 1011000 n Theorem: Let g: V n ⎯ 2 ⎯→ S be a ring homomorphism with a, b ∈ V 2 . Ker(c) + a = Ker (c) + b. Iff a + b ∈ Ker(c). Proof: If Ker(c) + a = Ker(c) + b, then a = 0 + a ∈ Ker(c) + a = Ker(c) + b Hence f K ∈ Ker(c) such that a = K + b ⇒ a+b = K ∈ Ker(c) Conversely, If a + b ∈ Ker(c), then a =(a+b) +b ∈ Ker(c) + b Since a + Ker(c) and b + Ker(c) are either disjoin are equal.Therefore Ker(c) + a = Ker(c) + b. Color decoding Let “C” be a linear code and assume that code word v ∈ C is transmitted and word w is received, resulting in error pattern e= v ⊕ w then w ⊕ e = v∈C. The above theorem suggests that the error pattern e and the received word w has the same color, Since error patterns of small weight are most likely to occur here is how MLD works for a linear code C. Color decoding algorithm i. ii. iii. iv. Let w ∈ Z2 be a received word find its color, let it is i. Build the set Gi of all words with color i. Trace a word e of least weight in G i . Then e ⊕ w is the code transmitted. n 583 Shoaib U Din and Khalil Ahmad E.R.Berlekamp and J.H.Conway , Winning ways for your mathematical plays. Academic Press, 1982. R.A Brualdi and V. Pless, “Greedy Codes”, JCT (A) 64 (1993), 10-30. J.H. Conway, “Integral Lexicographic Codes”, Discrete Mathematics 83 (1990) 219-235. L.Monroe, “Binary Greedy Codes”, Congressus Numerantium, vol. 104(1994), 49-63. W.Chen and T. Kløve, “On the second greedy weight for linear codes of dimension 3” Discrete Math. vol. 241, pp. 171-187, 2001 W. Chen and T. Kløve, Weight hierarchies of binary linear codes of dimension 4, Discrete Mathematics 238 (2001), 27-34. Ari Trachtenberg, Alexander Vardy,Lexicographic codes, “Lexicographic Codes” Proc.\ 31st Annual Conference on Information Sciences and Systems Ahmad khalil “g-value Decoding of Greedy Codes” Proc.\1st International Conference on Information and Com. Tech. IEEE Catalog No.05EX1176 pp. 99-100.(2005) W.C. Huffman and V. 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