Outline The Codes Bounds on the minimum distance Dimension of Graph Codes Special results when q is a square Performance Graph Codes Tom Høholdt, Jørn Justesen and Fernando Piñero The Technical University of Denmark Mumbay December 2013 Tom Høholdt, Jørn Justesen and Fernando Piñero The Technical University of Denmark Graph Codes
Outline The Codes Bounds on the minimum distance Dimension of Graph Codes Special results when q is a square Performance Product Codes
Code Construction The bipartite graph G is used to define a code over F q by associating a symbol with each edge and letting all edges that meet in a left vertex satisfy the parity checks of an ( n , k 1 , d 1 ) code C 1 and edges that meet in a right vertex satisfy the parity checks of an ( n , k 2 , d 2 ) code C 2 . The resulting code is denoted by ( G , C 1 : C 2 ) . Thus the length of the code is N = mn
Dimension K ≥ N − m ( n − k 1 ) − m ( n − k 2 ) so R ≥ r 1 + r 2 − 1
Adjacency Matrix The bounds on the minimum distance involve the second largest eigenvalue of the adjacency matrix of the graph. Let x 1 , x 2 , . . . , x m be the left vertices V 1 and y 1 , y 2 , . . . , y m the right vertices V 2 . Define the matrix M = m ij where � 1 if x i is connected to y j m ij = 0 else The adjacency matrix of the bipartite graph is then � � 0 M A = M T 0
Previous Results Sipser and Spielman 1996 If d 1 = d 2 = d D ≥ dm d − λ n − λ Janwa and Lal 2003 If d 1 ≥ d 2 ≥ λ 2 D ≥ m n ( d 1 d 2 − λ 2 ( d 1 + d 2 )) Roth and Skachek 2006 d 1 d 2 − λ √ d 1 d 2 D ≥ m n − λ
New Results I d 2 − λβ D ≥ md 1 n − λβ where � λ 2 ( d 1 − d 2 ) 2 + 4 d 1 d 2 ( n − d 1 )( n − d 2 ) λ ( d 1 − d 2 ) + β = 2 d 1 ( n − d 2 )
The fundamental observation is Lemma Let v ∈ R 2 m where v ⊥ 1 then < v , Av > ≤ λ < v , v > where λ is the second largest eigenvalue of A.
Proof Let ` E be a set of edges in G corresponding to the nonzero positions of a nonzero codeword of C . Let S be the subset of vertices in V 1 incident with ` E and let T be the subset of vertices in V 2 incident with ` E . We will get the bound on D from a bound of | ` E | . Suppose that | S | = a and | T | = α a , α ≥ 1, and let e be the average valency of the vertices in S , thus e α the average valency in T . Let v = ( v i ) be a vector of length 2 m where 1 if i ∈ S a − if i ∈ V 1 \ S m − a v i = β if i ∈ T − αβ a if i ∈ V 2 \ T m − α a where 0 < β ≤ 1.
Proof Let ` E be a set of edges in G corresponding to the nonzero positions of a nonzero codeword of C . Let S be the subset of vertices in V 1 incident with ` E and let T be the subset of vertices in V 2 incident with ` E . We will get the bound on D from a bound of | ` E | . Suppose that | S | = a and | T | = α a , α ≥ 1, and let e be the average valency of the vertices in S , thus e α the average valency in T . Let v = ( v i ) be a vector of length 2 m where 1 if i ∈ S a − if i ∈ V 1 \ S m − a v i = β if i ∈ T − αβ a if i ∈ V 2 \ T m − α a where 0 < β ≤ 1.
Proof Let ` E be a set of edges in G corresponding to the nonzero positions of a nonzero codeword of C . Let S be the subset of vertices in V 1 incident with ` E and let T be the subset of vertices in V 2 incident with ` E . We will get the bound on D from a bound of | ` E | . Suppose that | S | = a and | T | = α a , α ≥ 1, and let e be the average valency of the vertices in S , thus e α the average valency in T . Let v = ( v i ) be a vector of length 2 m where 1 if i ∈ S a − if i ∈ V 1 \ S m − a v i = β if i ∈ T − αβ a if i ∈ V 2 \ T m − α a where 0 < β ≤ 1.
Proof We can directly calculate 2 ma β v T Av = ( m − a )( m − α a ) ( me − na α ) The fundamental inequality and this result give 2 ma β ( m − a )( m − α a ) ( me − na α ) ≤ ( m − a ) 2 + a αβ 2 + ( m − α a ) α 2 β 2 a 2 a 2 λ ( a + ( m − a ) ( m − α a ) 2 ) and this by a straightforward calculation leads to the following bound on a 2 e β − λ ( 1 + αβ 2 ) a ≥ m (1) α 2 β n − λ ( 1 + β 2 ) which holds for any positive β .
Proof The lower bound on a is met if and only if v is an eigenvector associated with the eigenvalue λ , i.e. Av = λ v , and a necessary condition for this, where we only look at the upper part of A is a αβ βλ = e α − ( n − e a λ = e β − ( n − e ) α ) and m − a α m − a These two conditions lead to the following expressions for a e β − λ a = m (2) α ( β n − λ ) e α − βλ a = m (3) n − βλ and by eliminating a we get the equation for β . It can be seen that there is a positive solution less than 1.
Proof The lower bound on a is met if and only if v is an eigenvector associated with the eigenvalue λ , i.e. Av = λ v , and a necessary condition for this, where we only look at the upper part of A is a αβ βλ = e α − ( n − e a λ = e β − ( n − e ) α ) and m − a α m − a These two conditions lead to the following expressions for a e β − λ a = m (2) α ( β n − λ ) e α − βλ a = m (3) n − βλ and by eliminating a we get the equation for β . It can be seen that there is a positive solution less than 1.
Proof Maximizing the right side of (1) with respect to β actually leads to the same equation. Thus this is the sharpest lower bound that can be obtained by this method. The bound can be met if there is a subgraph on S and T with exactly valencies e and e α (which we expect will rarely be the case). Since D ≥ ea the lower bound increases with a and e , we thus get a new lower bound by choosing α = e d 2 since the bound on a decreases with α and then choosing e = d 1 . �
Proof Maximizing the right side of (1) with respect to β actually leads to the same equation. Thus this is the sharpest lower bound that can be obtained by this method. The bound can be met if there is a subgraph on S and T with exactly valencies e and e α (which we expect will rarely be the case). Since D ≥ ea the lower bound increases with a and e , we thus get a new lower bound by choosing α = e d 2 since the bound on a decreases with α and then choosing e = d 1 . �
Comparisons If n = 16 , λ = 4 , d 1 = 8 , d 2 = 4 the new bound gives D ≥ 4 m 5 Janwa-Lal gives D ≥ m 2 and Roth-Skachek D ≥ 0 . 78 m
New Results II By a more careful analysis we get If d 1 = d 2 = d D ≥ da where a ≤ n + 1 − d + ( n + 1 − d )( λ 2 − d 2 + 2 d − 1 ) m n + 1 + d 2 − d − λ 2
New Results II The graph of a generalized quadrangle over F 8 has m = 585 , n = 9 , λ = 4. With C 1 = C 2 = RS ext ( 9 , 6 , 4 ) The bound gives D ≥ 4 × 45 whereas the previous bounds give nothing.
A Question Since all the bounds are decreasing functions of λ it is natural to ask: How small can λ be?
An Answer A special case of a recent result ( Janwa +TH) Let G be a n -regular connected bipartite graph with m + m vertices then � mn − n 2 λ ≥ r − 1 where r is the rank of the transfer-matrix M and equality holds iff the eigenvalues of A are ± n with multiplicity 1, ± λ with multiplicity r − 1 and 0 with multiplicity 2 m − 2 r .
An Answer A special case of a recent result ( Janwa +TH) Let G be a n -regular connected bipartite graph with m + m vertices then � mn − n 2 λ ≥ r − 1 where r is the rank of the transfer-matrix M and equality holds iff the eigenvalues of A are ± n with multiplicity 1, ± λ with multiplicity r − 1 and 0 with multiplicity 2 m − 2 r .
Duals of Graph Codes Theorem ( G , C 1 : C 2 ) = ( G , C 1 : F n q ) ∩ ( G , F n q : C 2 ) ( G , C 1 : C 2 ) ⊥ = ( G , C ⊥ 1 : F n q ) + ( G , F n q : C ⊥ 2 ) . ( G , C 1 : F n q ) is the Graph Code on G where the codeword depends only on C 1 and V 1 . ( G , F n q : C 2 ) is the Graph Code on G where the codeword depends only on C 2 and V 2 . Their intersection is ( G , C 1 : C 2 ) . q ) ⊥ = ( G , C ⊥ ( G , C 1 : F n 1 : F n q ) implies the second statement.
Duals of Graph Codes Theorem ( G , C 1 : C 2 ) = ( G , C 1 : F n q ) ∩ ( G , F n q : C 2 ) ( G , C 1 : C 2 ) ⊥ = ( G , C ⊥ 1 : F n q ) + ( G , F n q : C ⊥ 2 ) . ( G , C 1 : F n q ) is the Graph Code on G where the codeword depends only on C 1 and V 1 . ( G , F n q : C 2 ) is the Graph Code on G where the codeword depends only on C 2 and V 2 . Their intersection is ( G , C 1 : C 2 ) . q ) ⊥ = ( G , C ⊥ ( G , C 1 : F n 1 : F n q ) implies the second statement.
Duals of Graph Codes Theorem ( G , C 1 : C 2 ) = ( G , C 1 : F n q ) ∩ ( G , F n q : C 2 ) ( G , C 1 : C 2 ) ⊥ = ( G , C ⊥ 1 : F n q ) + ( G , F n q : C ⊥ 2 ) . This theorem also improves the usual bound on the dimension of Graph codes. The usual bound counts checks given by ( G , C ⊥ 1 : F n q ) and ( G , F n q : C ⊥ 2 ) . We improve it by counting the dependent checks in common: ( G , C ⊥ 1 : C ⊥ 2 ) .
Properties of Graph Codes Let G be an n –regular, bipartite graph. Let C be a [ n , k , d ] code. Let γ G be the ratio of the second largest and largest eigenvalues of (the adjacency matrix of) G , then d n − γ G d ( G , C : C ) ≥ | E | d n 1 − γ G ( G , C : C ) can be decoded in linear time.
A Problem But determining the dimension of Graph Based codes remains difficult.
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