Generalized Sphere Packing Bound Arman Arman Fazeli Fazeli (1) , Alexander , Alexander Vardy Vardy (1) , and , and Eitan Eitan Yaakobi Yaakobi (2) (1) - University of (2) - Technion California San Diego Israel Institute of Technology 1
The Sphere Packing Bound • Upper bound on a code C with min dist 2r+1 – • This bound is valid for other cases as well where the error graph is regular ( ) what happens if the graph is not regular? – Naïve solution: choose to be the minimum size of a ball in the graph 2
What about other bounds? • The Gilbert-Varshamov lower bound: There exists a code with min dist r+1 of size • If the graph is not regular, it is possible to choose as the average size of a ball • There exists a code with min dist r+1 of size Does the same analogy hold for the sphere packing bound? Is an upper bound on a code with min dist 2r+1 ? 3
The Deletion Channel • An example of non-regular graph – 10010 -> 0010, 1010, 1000, 1001 – 11100 -> 1100, 1110 – 10101 -> 0101, 1101, 1001, 1011, 1010 • It is not possible to apply the sphere packing bound • Previous results – Levenshtein ‘66 : asymptotic upper bound – Kulkarni & Kiyavash ‘12 : a method to derive explicit non- asymptotic upper bound using tools from hypergraph theory • Can this method be generalized for other graphs? – Grain errors: • Kashyap & Zemor ‘13; Gabrys, Yaakobi, Dolecek ‘13 – Multi- permutations with the Kendall’s tau dist • Buzaglo, Yaakobi, Etzion, Bruck ‘13 4
Hypergraphs Let H=(X,E) be a hypergraph , where • – X={x 1 ,…,x n } – set of vertices, E={E 1 ,…,E m } – set of hyperedges – A is a binary n × m incidence matrix of H • Matching - a collection of pairwise disjoint hyperedges – The matching number ν ( H ) - the size of the largest matching • Transversal - a vertices subset that intersects every hyperedge – The transversal number τ (H) - the size of the smallest transversal 5
Hypergraphs • The matching number • The transversal number • These problems satisfy weak duality ν ( H ) ≤ τ (H) • The relaxation versions of these problems satisfy strong duality • Every vector w in τ *(H) is called a fractional transversal 6
The Deletion Channel – KK’12 • Define a hypergraph H(X,E): – X = {0,1} n-1 , E = {all 2 n single-deletion balls} • Every single-deletion correcting code of length n is a matching in the hypergraph H • Find the value of τ *(H) or any fractional transversal to get an explicit upper bound 7
The General Case G=(X,E) is a graph describing an error channel graph • – X = the set of all possible words (transmitted and received) – E = the set of vertices pairs of dist one • The distance d(x,y) b/w x and y is the length of the shortest path from x to y (not necessarily symmetric) – B r (x) = {y ∊ X : d(x,y) ≤ r}; deg r (x) = |B r (x)| For any r>0, H(G,r)=(X r ,E r ) is a hypergraph for G • – X r =X , E r ={B r (x) : x ∊ X} Every code C in G is a matching in H(G,r) • A G (n,d) - the max size of a code w/ min dist d in G • For every r>0: • Q : Does the following hold? is called the Average Sphere Packing Value : ASPV(G,r) 8
Example: The Z Channel • G Z =(X Z ,E Z ), X Z ={0,1} n B Z,1 (10010)={10010,00010,10000} • H(G Z,r ) = (X Z,r ,E Z,r ), X Z,r =X Z ={0,1} n 9
Example: The Z Channel • The average size of a ball with radius r • The average sphere packing value • For r=1: 10
So what is the problem? • A channel graph G=(X,E) w/ hypergraph H(G,r)=(X r ,E r ) • A code with min dist 2r+1 in G is a matching in H(G,r) • An upper bound is given by • The good news : there is an explicit upper bound! • The bad news : It is not necessarily easy to calculate it – Need to solve a linear programming… – Usually the number of variables and constraints in exponential • Our goal : how to calculate the value of 11
Some General Results • Lemma : If for all x ∊ X, deg r in (x) ≤ Δ then Proof : Since deg r in (x) ≤ Δ then ⇒ For any transversal w , Therefore, 12
Some General Results • Lemma : If for all x ∊ X, deg r in (x) ≤ Δ then • Lemma : If for all x ∊ X, deg r (x) ≥ Δ , then Proof : The vector w = 1 / Δ is a fractional transversal Thus, 13
Some General Results • Lemma : If for all x ∊ X, deg r in (x) ≤ Δ then • Lemma : If for all x ∊ X, deg r (x) ≥ Δ , then • Corollary : If G is symmetric and regular then the generalized sphere packing bound and the sphere packing bound coincide, and 14
Monotonicity and Fractional Transversals • The vector w is a fractional transversal if w ≥ 0 and • Lemma : The vector w , given by is a fractional transversal Proof : If y ∊ B r (x i ), then x i ∊ B r in (y) and Therefore, 15
Monotonicity and Fractional Transversals • The vector w is a fractional transversal if w ≥ 0 and • Lemma : The vector w , given by is a fractional transversal • Def : G is called monotone if for all x ∊ X and y ∊ B r (x) • Lemma : If G is monotone then the vector is a fractional transversal • Corollary : If G is monotone an upper bound on A G (n,2r+1) is called the monotonicity upper bound MB(G,r) 16
Cont. Ex: The Z Channel • x,y ∊ {0,1} n , if y ∊ B Z,r (x) , w H (y) ≤ w H (x) and deg r (y) ≤ deg r (x) • The vector given by is a fractional transversal • The monotonicity upper bound for the Z channel: • For r=1: • The average sphere packing bound • Is it possible to do better…? 17
Can we find the optimal transversal? • w is a fractional transversal if w ≥ 0 and • For the Z channel: – 2 n constraints: Ex, n=3: • 111 : w 111 +w 110 +w 101 +w 011 ≥ 1 • 110 : w 110 +w 100 +w 010 ≥ 1, 101 : w 101 +w 100 +w 100 ≥ 1, 011 : w 011 +w 010 +w 001 ≥ 1 • 100 : w 100 +w 000 ≥ 1, 010 : w 010 +w 000 ≥ 1, 001 : w 001 +w 000 ≥ 1 • 000 : w 000 ≥ 1 – Probably vectors w/ the same weight will have the same value – If so, only n+1 constraints: • 3 : w 3 +3w 2 ≥ 1 • 2 : w 2 +2w 1 ≥ 1 • 1 : w 1 +w 0 ≥ 1 • 0 : w 0 ≥ 1 • Is it still possible to find the value of ? 18
Automorphisms on Graphs • Given a graph G=(X,E) , an automorphism is a permutation that preserves adjacency π :X->X s.t. (x,y) ∊ E iff ( π (x), π (y)) ∊ E • The automorphisms set Aut(G) ={ π ∊ S n : π is an automorphism in G} is a subgroup of S n under composition • Aut(G) induces an equivalence order R on X: (x,y) ∊ R iff there exists π ∊ Aut(G) and π (x)=y and partitions X into n(G) equivalence classes • For a vector w and automorphism π, the vector w π is ( w π ) i = w π( i) • Lemma : If w is a transversal and π an automorphism then w π is a transversal as well 19
Automorphisms on Graphs • Given a graph G=(X,E) , an automorphism is a permutation that preserves adjacency π :X->X s.t. (x,y) ∊ E iff ( π (x), π (y)) ∊ E Proof : • The automorphisms set Need to show: for 1 ≤ i ≤ n Aut(G) ={ π ∊ S n : π is an automorphism in G} y ∊ B r (x i ) iff π(y) ∊ B r ( π( x i )) is a subgroup of S n under composition Therefore, • Aut(G) induces an equivalence order R on X: (x,y) ∊ R iff there exists π ∊ Aut(G) and π (x)=y and partitions X into n(G) equivalence classes • For a vector w and automorphism π, the vector w π is ( w π ) i = w π( i) • Lemma : If w is a transversal and π an automorphism then w π is a transversal as well 20
Automorphisms on Graphs • Aut(G) = { π ∊ S n : π is an automorphism in G} • An equivalence order R on X: (x,y) ∊ R iff there exists π ∊ Aut(G) and π (x)=y • Lemma : If w is a transversal and π an automorphism then w π is a transversal as well • W c = { w : w is a transversal and Σ w i =c} • Theorem : If W c ≠ ∅ then W c contains a transversal which assigns the same weight to the equivalence classes of R • This result holds also for any subgroup of Aut(G) 21
Cont. Ex: The Z Channel • For every σ ∊ S n define a permutation π σ :{0,1} n ->{0,1} n for all x ∊ {0,1} n , ( π σ (x)) i =x σ (i) • The set K={ π σ : σ ∊ S n } is a subgroup of Aut(G Z ) and partitions {0,1} n into n+1 equivalence classes X K (G Z ) = {X 0 ,X 1 ,…,X n }, X i =all vectors of weight i • The generalized sphere packing bound now becomes: 22
Cont. Ex: The Z Channel • The generalized sphere packing bound now becomes: • For r=1 : where , and – For example, n=3: w * 3 =0, w * 2 = (1-0)/3=1/3, w * 1 = (1-1/3)/2=1/3, w * 0 = 1 23
Best known upper bound by Weber, De Vroedt, and Boekee ‘88 24
Cont. Ex: The Z Channel • For arbitrary r, the optimal solution is given by 25
Comparison for r=2 Comparison for r=3 26
Limited Magnitude Channels • Asymmetric errors G A,q =(X,E): X=[q] n • The graph is monotone • The monotonicity upper bound for r=1 is • The average sphere packing bound 27
The Asymmetric Channel • The linear programming problem to find is given by • By automorphisms as in the Z channel we can divide to the equivalence classes 28
The Asymmetric Channel • Theorem : the vector given by is a fractional transversal 29
Results for q=3 30
The Symmetric Channel • No longer monotonicity because the channel is symmetric • Can follow similar steps… Comparison for q=3 Comparison for q=4 31
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