aa r X i v : . [ m a t h . C O ] J a n A New Formula for the Minimum Distance of anExpander Code
Sudipta Mallik
Department of Mathematics and Statistics, Northern Arizona University, 801 S. Osborne Dr.PO Box: 5717, Flagstaff, AZ 86011, [email protected]
January 6, 2021
Abstract
An expander code is a binary linear code whose parity-check matrix is the bi-adjacency matrix of a bipartite expander graph. We provide a new formula for theminimum distance of such codes. We also provide a new proof of the result that2(1 − ε ) γn is a lower bound of the minimum distance of the expander code given by a( m, n, d, γ, − ε ) expander bipartite graph. Binary linear codes can be constructed from graphs. One such construction was given frombipartite graphs by Tanner in [6]. Sipser and Spielman constructed expander codes frombipartite expander graphs in [5]. One of the goals of all these constructions was to havelinear codes with relatively large minimum distance for efficient error correction. For moredetails on the literature of linear codes and bipartite graphs, see [1, 2, 5, 7]. In this articlewe provide a new formula for the minimum distance of expander codes. We also provide anew proof of the result that 2(1 − ε ) γn is a lower bound of the minimum distance of theexpander code given by a ( m, n, d, γ, − ε ) expander bipartite graph.Now we present a brief introduction to coding theory: A binary linear code C of length n and dimension k is a k dimensional subspace of F n where F is the binary field. The code C is called an [ n, k ]-code. The support of a codeword x ∈ C is the set of indices i such that i th entry of x is 1. The Hamming weight w H ( x ) of a vector x ∈ F n is the size of the supportof x . The Hamming distance , denoted by d H ( x , y ), between two codewords x and y in C is d H ( x , y ) = w H ( x − y ). The minimum distance of C , denoted by d ( C ), is the minimumdistance between distinct codewords in C . Note that d ( C ) is the minimum Hamming weightof a nonzero codeword in C . We call C to be an [ n, k, d ] code when d ( C ) = d . A binarymatrix H is called the parity-check matrix of C if C the null space of H , i.e., C = { c ∈ F n | H c T = 0 } . L R
Figure 1: A (5 , , , , ) expander graph.The minimum distance d ( C ) can be expressed as the minimum number of linear depen-dent columns of the parity-check matrix of C as follows: Theorem 1.1. [4, Theorem 2.2] Let C be a linear code and H its parity-check matrix. Then C has minimum distance d if and only if any d − columns of H are linearly independentand some d columns of H are linearly dependent. For a vertex v of a graph G , the set of all vertices in G adjacent to v is called the neighbor of v , denoted by N( v ). For a set S of vertices of G , N( S ) denotes the union of neighbors ofvertices in S . Now we define a bipartite expander graph based on its definition in [5, 6] withthe roles of left and right set of vertices switched: Definition 1.2.
Suppose G is a bipartite graph with vertex set L ·∪ R such that | L | = m , | R | = n , each edge of G joins a vertex of L with a vertex of R , and each vertex of R is adjacentto exactly d vertices of L . For positive γ and α , G is called a ( m, n, d, γ, α ) expander graph if for each set S ⊆ R satisfying | S | ≤ γn , we have | N ( S ) | ≥ dα | S | . Example 1.3.
The bipartite graph in Figure 1 is a (5 , , , , ) expander graph. Eachvertex in R has degree d = 2. If S ⊆ R satisfies | S | ≤ γn = 2, then | S | = 1 or 2. For | S | = 1, | N ( S ) | = 2 ≥ = dα | S | . Also for | S | = 2, | N ( S ) | ≥ = dα | S | . Definition 1.4.
Suppose G is a ( m, n, d, γ, α ) expander graph and B is the m × n bi-adjacency matrix of G , i.e., A = (cid:20) O m BB T O n (cid:21) is the adjacency matrix of G . The binary linear code whose parity-check matrix is B is calledthe expander code of G , denoted by C ( G ). In other words, C ( G ) = { c ∈ F n | B c T = in F } . xample 1.5. The bi-adjacency matrix of the (5 , , , , ) expander graph G in Figure 1is given by B = . The expander code C ( G ) of G is given by C ( G ) = { c ∈ F | B c T = in F } . We start with the following notation and definition of von from [3].
Definition 2.1.
For a nonempty subset S of vertices of a graph G , the set of vertices of G with odd number of neighbors in S is denoted by von( S ), i.e.,von( S ) = { v ∈ V ( G ) : | N( v ) ∩ S | is odd } . Example 2.2.
Consider the (5 , , , , ) expander graph G in Figure 1. For v = 6 , , , { v } ) = N( v ). For S = { , , } , von( S ) = { , , , } . For S = { , , } , von( S ) = ∅ .Now we proceed to the main results of this article which give a new formula of theminimum distance of an expander code. Theorem 2.3.
Suppose G is a bipartite graph with vertex set L ·∪ R such that | L | = m , | R | = n , and B is the m × n bi-adjacency matrix of G . Let S be a nonempty subset of R . If von( S ) = ∅ , then the columns of B indexed by S are linearly dependent. Conversely if thecolumns of B indexed by S are minimally linearly dependent, then von( S ) = ∅ .Proof. Suppose von( S ) = ∅ where S = { i , i , . . . , i t } ⊆ R . Then B i + B i + · · · + B i t ≡ B i , B i , . . . , B i t of B are linearly dependent.Conversely suppose S = { , , . . . , k } ⊆ R and B , B , . . . , B k are minimally linearlydependent columns of B . Then B + B + · · · + B k ≡ S ) = ∅ .Otherwise let i ∈ von( S ). Then( B + B + · · · + B k ) i ≡ , a contradiction to B + B + · · · + B k ≡ S ) = ∅ . Theorem 2.4.
Suppose G is a bipartite graph with vertex set L ·∪ R such that | L | = m , | R | = n , and B is the m × n bi-adjacency matrix of G . Suppose C is the binary linear codewhose parity-check matrix is B . Then the minimum distance d ( C ) of C is given by d ( C ) = min {| S | : ∅ = S ⊆ R, von( S ) = ∅ } . roof. First note that B is the parity-check matrix of C . By Theorem 1.1, the support of acode word in C with weight d ( C ) is the set of indices of some minimally dependent columnsof B , say indexed by T for some nonempty subset T of R . By Theorem 2.3, von( T ) = ∅ .Then d ( C ) = | T | ≥ min {| S | : ∅ = S ⊆ R, von( S ) = ∅ } . To show the equality, on the contrary suppose there is a nonempty subset S of R for which d ( C ) > | S | and von( S ) = ∅ . Then by Theorem 2.3, we find | S | linearly dependent columnsof B giving a codeword of C with weight less than d ( C ), a contradiction. Example 2.5.
Consider the (5 , , , , ) expander graph G in Figure 1. Suppose C is thebinary linear code whose parity-check matrix is the bi-adjacency matrix of G . We can verifythat for any nonempty set S ⊆ R with | S | ≤
2, we have von( S ) = ∅ . Now for S = { , , } ,von( S ) = ∅ . Thus by Theorem 2.4, d ( C ) = min {| S | : ∅ = S ⊆ R, von( S ) = ∅ } = |{ , , }| = 3 . The preceding theorem results in a new formula for the minimum distance of expandercodes.
Theorem 2.6.
Suppose G is a ( m, n, d, γ, α ) expander graph with vertex set L ·∪ R such that | L | = m and | R | = n . Then the minimum distance d ( C ) of the expander code C of G isgiven by d ( C ) = min {| S | : ∅ = S ⊆ R, von( S ) = ∅ } . Using the minimum distance formula given in Theorem 2.6, we provide a new proof of thefollowing known result which gives a lower bound of the minimum distance of an expandercode.
Theorem 2.7.
Let < ε < and γ > such that γn is a positive integer. Suppose G is a ( m, n, d, γ, − ε ) expander graph with vertex set L ·∪ R such that | L | = m and | R | = n . Thenthe minimum distance d ( C ) of the expander code C of G has the following lower bound: d ( C ) ≥ − ε ) γn. To prove Theorem 2.7, we first prove the following lemmas:
Lemma 2.8.
Let < ε < . Suppose G is a ( m, n, d, γ, − ε ) expander graph with vertexset L ·∪ R such that | L | = m and | R | = n . For each set S ⊆ R satisfying | S | ≤ γn , we have d (1 − ε ) | S | ≤ | von( S ) | ≤ | N( S ) | . Proof.
Suppose S ⊆ R satisfies | S | ≤ γn . The second inequality follows from the factvon( S ) ⊆ N( S ) ⊆ L by definition. To show the first inequality, note that there are d | S | edges between vertices in S and vertices in N( S ) ⊆ L and each vertex in von( S ) has atleast one neighbor in S . Also each vertex in N( S ) \ von( S ) has even number (at least 2) ofneighbors in S . Thus d | S | ≥ | von( S ) | + 2 | N( S ) \ von( S ) | = 2 | N( S ) | − | von( S ) | | von( S ) | ≥ | N( S ) | − d | S | . Since | S | ≤ γn and G is a ( m, n, d, γ, − ε ) expander graph, | N( S ) | ≥ d (1 − ε ) | S | . Thus | von( S ) | ≥ | N( S ) | − d | S | ≥ d (1 − ε ) | S | − d | S | = d (1 − ε ) | S | . Lemma 2.9.
Suppose G is a bipartite graph with vertex set L ·∪ R . Let A and B be nonemptydisjoint subsets of S ⊆ R such that S = A ∪ B . If von( S ) = ∅ , then von( A ) = von( B ) .Proof. Let von( S ) = ∅ . To show von( A ) ⊆ von( B ), suppose x ∈ von( A ). We claim x ∈ von( B ). Otherwise x / ∈ von( B ), i.e., x is adjacent to an even number of vertices in B .Since x ∈ von( A ), x is adjacent to an odd number of vertices in A . Thus x is adjacent toan odd number of vertices in S = A ∪ B . Therefore von( S ) = ∅ , a contradiction. Thusvon( A ) ⊆ von( B ). Similarly we can show that von( B ) ⊆ von( A ).Using the above lemmas, we prove Theorem 2.7. Proof of Theorem 2.7.
By Theorem 2.6, consider a nonempty set S ⊆ R such that d ( C ) = | S | and von( S ) = ∅ . To prove by contradiction, suppose 2(1 − ε ) γn > d ( C ) = | S | .Case 1. | S | ≤ γn By Lemma 2.8, d (1 − ε ) | S | ≤ | von( S ) | . Since ε < , we have0 < d (1 − ε ) | S | ≤ | von( S ) | , which implies von( S ) = ∅ , a contradiction.Case 2. | S | > γn In this case 2(1 − ε ) γn > | S | > γn. Choose a nonempty subset T of S ⊆ R such that | T | = γn . Then by Lemma 2.8, d (1 − ε ) γn = d (1 − ε ) | T | ≤ | von( T ) | ≤ | N( T ) | . (2.1)Note that | S \ T | = | S | − | T | < − ε ) γn − γn = (1 − ε ) γn. Since each vertex in S \ T has d neighbors in L , by Lemma 2.8, | von( S \ T ) | ≤ | N( S \ T ) | ≤ d | S \ T | < d (1 − ε ) γn. (2.2)Combining (2.1) and (2.2), we have | von( S \ T ) | < d (1 − ε ) γn ≤ | von( T ) | , which implies von( S \ T ) = von( T ). Since S = S ∪ ( S \ T ) and von( S ) = ∅ , by Lemma 2.9,we have von( S \ T ) = von( T ), a contradiction.5 bservation 2.10. If we like to find the minimum distance d ( C ) of the expander code C of a ( m, n, d, γ, − ε ) expander graph G with vertex set L ·∪ R by brute force using Theorem2.6, then we need to consider all possible subset S ⊆ R such that von( S ) = ∅ . But becauseof Theorem 2.7, we need to look at only S ⊆ R satisfying | S | > − ε ) γn . Example 2.11.
Consider the expander code C ( G ) of the (5 , , , , ) expander graph G inFigure 1. Note that 1 − ε = . By Theorem 2.7, we need to look at only S ⊆ R satisfying | S | > − ε ) γn = . So we look at nonempty sets S ⊆ R satisfying | S | ≥ S ) = ∅ . For S = { , , } , von( S ) = ∅ . Thus by Theorem 2.6, d ( C ( G )) = min {| S | : ∅ = S ⊆ R, von( S ) = ∅ } = |{ , , }| = 3 . Acknowledgments
The author would like to thank his colleague Dr. Bahattin Yildiz for his valuable suggestions.
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