Capacity of Hexagonal Checkerboard Codes
1
Capacity of Hexagonal Checkerboard Codes Zhun Deng, Jie Ding, Mohammad Noshad, and Vahid Tarokh
Abstract In this paper, we propose a new method to bound the capacity of checkerboard codes on the hexagonal lattice. This produces rigorous bounds that are tighter than those commonly known.
arXiv:1506.02163v1 [cs.IT] 6 Jun 2015
Index Terms Capacity, constrained codes, hexagonal lattice, transfer matrix.
I. I NTRODUCTION HECKERBOARD codes are two-dimensional binary codes that are designed to satisfy specific constraints [1], [2]. An example is the two-dimensional rectangular binary arrays that satisfy the (d, k) run-length constraint— there are at most k 0’s and the number of 0’s between any neighboring 1’s is at least d in each row and column [3]–[7]. It may naturally arise from data storage on a surface [8], [9]. An example of the two dimensional (1, ∞) run-length constraint is shown in Fig. 1. In practice, the distance between two data recorded points on a recording device should be no less than a given threshold due to the physical and fidelity constraints. It is of interest to study the checkerboard code on the hexagonal lattice, because it is known that the lattice arrangement of circles with the highest density in two dimensional Euclidean space is the hexagonal packing arrangement, in which the centers of the circles are arranged on the hexagonal lattice [10]. Specifically, we study the capacity of the checkerboard code shown in Fig. 2, where only 0’s can be arranged in the six neighbors of any 1, while both 0 and 1 can be arranged in the six neighbors of any 0 (which is referred to as the hexagonal constraint). ∞ S S i = A2 . Let S1 ⊂ S2 ⊂ · · · be a sequence of finite subsets of two dimensional hexagonal lattice A2 such that
C
i=1
Let f (Si ) (i ∈ N) be the total number of distinct arrangements of 0’s and 1’s satisfying the hexagonal constraint given a two-dimensional region Si , and |Si | denotes the number of points inside Si . The capacity of the hexagonal checkerboard code is defined as log2 f (Si ) , (1) C ∗ = lim i→∞ |Si | if the limit exists. The subsets Si , i = 1, 2, · · · , considered in this work will mainly be the zigzag regions shown in Fig. 3. The existence of the capacity has been established in some prior papers [11], [12] using the subadditivity property of a double sequence. Weeks et al. [1] studied a similar problem on the integer lattice. They provided rigorous lower and upper bounds on the capacity and produced high precision approximations of using Richardson extrapolation. A detailed discussion on their results is included in Section V. Wilf [13] proposed the transfer matrix method to study the problem of arranging kings in a checkerboard. Calkin et al. [14] further used the transfer matrix to compute the bounds on the capacity of the diamond constraint shown in Fig. 1. However, their method cannot be directly used in calculating the capacity of hexagonal lattice. Inspired by Calkin et al.’s work, we propose a modified transfer matrix method that provides rigorous bounds on the capacity of the hexagonal constraint. The bounds presented in this work are tighter than those calculated by Weeks et al. [1], under the same computational complexity. II. T RANSFER M ATRIX
FOR
H EXAGONAL L ATTICE A RRAYS
A. Notations Let {Fi }∞ i=0 denote the Fibonacci numbers, i.e., F0 = 1, F1 = 1, and Fi = Fi−1 + Fi−2 . For any real-valued matrix A, let σmax (A), AT , and Trace(A) respectively denote the largest absolute value of the eigenvalues, the transpose, and the sum of all the diagonal elements of A. We represent the inner product of any two real-valued vectors x and y of the same size by hx, yi. The notations 1s×r and 0s×r respectively denote the s × r matrices of Z. Deng, J. Ding, M. Noshad and V. Tarokh are with the School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA. E-mail: [email protected] , [email protected] , [email protected] , [email protected] .
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Fig. 1: Illustration of the diamond and square constraints
Fig. 2: Illustration of the hexagonal constraint all ones and all zeros. We drop the subscripts s × r when there is no ambiguity. We note that for any real-valued symmetric r × r matrix A and positive integer n, the following inequality holds (σmax (A))n ≥
h1r×1 , An 1r×1 i . h1r×1 , 1r×1 i
(2)
We provide the following lemma for the future convenience. It follows immediately from the Perron-Frobenius Theorem and Proposition 4.2.1 in [15]. Lemma 1. Let A be a real-valued r×r matrix that is nonnegative and irreducible. Then A has a positive eigenvector T v = [v[1], v[2], · · · , v[r]] with corresponding eigenvalue σmax (A). Besides this, the following inequality holds for any positive integer n rd e (σmax (A))n ≤ h1, An 1i ≤ (σmax (A))n (3) d e where e = min{v[1], v[2], · · · , v[r]} and d = max{v[1], v[2], · · · , v[r]}. B. Transfer Matrix Let Gm,n denote the zigzag shaped array on the hexagonal lattice shown in Fig. 3, where m is the number of points in each row and n is the number of rows. The row vectors are categorized into two different types: the ones whose first point lies on the left of its upper row (or lower row) are termed as “L” (left) type, while the others belong to “R” (right) type. Assume that the lowest row of Gm,n is of “R” type. We emphasize that the vector “100001“ of “L” type is different from “100001” of “R” type. Clearly, an “L” type vector cannot be attached to an “L” type vector, while an “R” type vector can be attached to an “L” type vector–either upwards or downwards (see Fig. 3).
Fig. 3: Illustration of the zigzag hexagonal array with m rows and n columns
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Let f (m, n) denote the total number of valid arrangements of 0’s and 1’s in Gm,n satisfying the hexagonal constraint. The value of (1) for the given set of of regions therefore becomes C=
lim
m,n→∞
log2 f (m, n) . mn
(4)
We will provide lower and upper bounds on C in the rest part of the paper. A row vector is called “valid” if none of its two neighboring elements are both ones. Let am denote the number of valid row vectors of length m. Then am = Fm+1 . In fact, if a valid row starts with element zero, the remaining m − 1 elements have am−1 valid arrangements; if the row starts with one, the second element must be zero and the remaining m − 2 elements have am−2 valid arrangements. So, we obtain am = am−1 + am−2 . Besides, it is easy m to see that a1 = 2, a2 = 3, which results in am = Fm+1 . Let Am = {ui }2a i=1 represent the collection of all the “L” type and “R” type valid row vectors of length m (without loss of generality, the “R” type are arranged before the m belong to “R” type). Clearly, each row of Gm,n belongs to Am . “L” type within Am , i.e., {ui }ai=1 Next, we define the transfer matrix Tm to be a 2am × 2am binary matrix whose (i, j)th entry is 1 if uj can be attached to ui (as is depicted in Fig. 3) Tm [i, j] = , 1 ≤ i, j ≤ 2am . (5) 0 otherwise Note that Tm is a symmetric matrix due to its construction. Let f (m, n, i) denote the number of valid arrangements in Gm,n whose top row is ui . Then we obtain f (m, n + 1, j) =
2a m X
f (m, n, i)Tm [i, j].
(6)
i=1
Let fm,n = [f (m, n, 1), f (m, n, 2), · · · , f (m, n, 2am)]. Therefore, (6) can be rewritten in the following form fm,n+1 = Tm fm,n , and fm,1 = [1, 1 · · · , 1, 0, 0, · · · , 0]T
(am 1’s) ,
from which we get
n−1 f (m, n) = 1, Tm fm,1 .
(7)
Due to symmetry, f (m, n) remains the same if the lowest row of Gm,n is “L” type instead of “R” type, i.e.,
n−1 f (m, n) = 1, Tm (1 − fm,1 ) ,
(8)
Combining (7) and (8) yields
n−1 2f (m, n) = 1, Tm 1 ,
which leads to the following lemma.
(9)
Lemma 2. For any given positive integer m, we obtain i h h
1 i 1 n−1 1, Tm 1 n lim log2 f (m, n) n lim log2 log2 f (m, n) log2 [σmax (Tm )] lim = n→∞ = n→∞ = . (10) n→∞ nm m m m Proof. Tm is an irreducible matrix since the directed graph it represents is strongly-connected. In fact, we can build a path from one vertex to any other vertex by inserting vertices that represent the all-zero vectors. Furthermore, Lemma 1 implies that there are positive constants d and e that depend on m such that
2am d e n−1 (σmax (Tm ))n−1 ≤ 1, Tm 1 ≤ (σmax (Tm ))n−1 . (11) d e Raising both sides of (11) to the power of 1/(n − 1) and letting n → ∞, we get
1/(n−1) n−1 lim 1, Tm 1 = σmax (Tm ) (12) n→∞
which implies the existence of the limit in (10). From Lemma 2 and since
lim [log2 f (m, n)]/mn exists, the iterated limit lim lim [log2 f (m, n)]/mn also
m,n→∞
m→∞ n→∞
exists and they are equal. Therefore, we only need to compute bounds on the iterated limit.
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Example 1. If m = 3, the valid rows are [0, 0, 0]R , [0, 0, 1]R , [0, 1, 0]R , [1, 0, 0]R , [1, 0, 1]R , [0, 0, 0]L , [0, 0, 1]L , [0, 1, 0]L , [1, 0, 0]L , [1, 0, 1]L , and the transfer matrix is 1 1 1 1 1 1 0 1 1 0 0 A T3 = , where A = 1 0 0 1 0 . AT 0 1 1 0 0 0 1 0 0 0 0 III. L OWER B OUNDS
ON
T HE C APACITY
We denote the shaded part of Gm,n that is formed by concatenating n column vectors by Gshaded m,n , as shown in n . Fig. 4. Similar to the arguments in Subsection II-B, there are an = Fn+1 valid column vectors, denoted by {wi }ai=1 We define Xn to be a an × an binary matrix whose (i, j)th entry is 1 if wj can be attached to wi , 1 ≤ i, j ≤ an . (13) Xn [i, j] = 0 otherwise Example 2. If n = 3, the valid columns are [0, 0, 0] , [0, 0, 1] , [0, 1, 0] , [1, 0, 0] , [1, 0, 1], and the transfer matrix is 1 1 1 1 1 1 0 0 1 0 X3 = 1 1 0 0 0 . 1 1 1 0 0 1 0 0 0 0
One can observe that each row of of Gshaded m,n is of length m − hn , where hn = [n/2] ([u] denotes the largest integer that is no larger than u). Similar to the derivation of (7), the
number of distinct arrangements satisfying the m−hn −1 hexagonal constraint in Gshaded is calculated to be 1, X 1 . Besides this, the number of points in Gm,n but m,n n is nm − n(m − h ) = nh . Thus, there exists a positive integer cm,n , which depends on m, n and not in Gshaded n n m,n upper-bounded by 2nhn , such that 1
1
n−1 1, Tm 1 = f (m, n) = cm,n 1, Xnm−hn −1 1 . (14) 2 2 For example, when n = 5 (Fig. 4), we have h5 = 2 and 1 ≤ c5 ≤ 210 .
Fig. 4: Illustration of the shaded part of Gm,n Because of (14) and the fact that Tm is symmetric, for any positive integer q we obtain m−h
(σmax (Tm ))n ≥ ≥
−1
n+2q+1 n+2q q n q 1i h1, Tm 1i cm,n+2q+1 h1, Xn+2q+1 hTm 1, Tm Tm 1i = = q q 2q m−h2q+1 −1 hTm 1, Tm 1i c m,2q+1 h1, Tm 1i h1, X2q+1 1i
1 2(2q+1)h2q+1
m−h
n+2q+1 h1, Xn+2q+1
−1
1i
m−h −1 h1, X2q+1 2q+1 1i
.
As a specific case, we choose n = 1, q = 8 and obtain σmax (Tm ) ≥
1 217h17
m−h18 −1 h1, X18 1i . m−h17 −1 h1, X17 1i
(15)
5
n 1 2 3 4 5 6
λmax 1.618 2.147 3.054 4.233 5.922 8.256
n 7 8 9 10 11 12
λmax 11.526 16.082 22.443 31.319 43.706 60.991
n 13 14 15 16 17 18
λmax 85.112 118.773 165.747 231.297 322.773 450.425
TABLE I: The largest eigenvalue of Xn
Fig. 5: Illustration of Bn Following a similar argument as in the proof of Lemma 2, Xn is irreducible and the following equality holds for any fixed n > 0
1 1/m σmax (Xn ) = lim h1, Xnm 1i = lim cm,n 1, Xnm−hn −1 1 m . (16) m→∞
m→∞
We thus obtain
log2 [σmax (Tm )] ≥ log2 lim C = lim m→∞ m→∞ m
1 217h17
m−h18 −1 h1, X18 1i m−h17 −1 h1, X17 1i
! m1
= log2
σmax (X18 ) σmax (X17 )
= 0.4807676144.
The largest eigenvalues of the transfer matrix Xn are listed in Table I for n = 1, 2, · · · , 18 . IV. U PPER B OUNDS
ON
T HE C APACITY
Since Tm is a real and symmetric matrix, the following inequality holds 1 σmax (Tm ) ≤ Trace(Tm )2n ) 2n ,
where
2n Trace(Tm )=
X
Tm [x0 , x1 ] Tm [x1 , x2 ] · · · Tm [x2n−1 , x0 ].
(17) (18)
1≤x0 ,x1 ,··· ,x2n−1 ≤2am
Clearly, the right hand side of (18) is the total number of valid arrangements of the row vectors (ux0 , ux1 , · · · , ux2n−1 , ux0 ) in a zigzag way from bottom to top, as shown in Fig. 5. We calculate the sum in (18) by considering the zigzag array as a concatenation of (2n + 1) × 1 valid column vectors (Fig. 5). The top element should be the same as the bottom one in each column, so that the first row and the last row become the same. We denote the n . The number of elements in Bn , denoted by bn , is F2n−2 + F2n collection of valid column vectors as Bn = {vi }bi=1 (n > 1) due to the following reason. If the bottom and top elements of a (2n + 1) × 1 column vector are 0, the remaining 2n − 1 elements have F2n valid arrangements; if the bottom and top elements are 1, the elements next to them must be 0, and the remaining 2n − 3 elements have F2n−2 valid arrangements. We define Bn to be a bn × bn binary matrix whose (i, j)th entry is 1 if vj can be attached to vi Bn [i, j] = , 1 ≤ i, j ≤ bn . (19) 0 otherwise
2n Using an equation similar to (6), we get Trace(Tm ) = 1, Bnm−1 1 , which leads to 1 h m1
1 i 2n 1 log2 [σmax (Bn )] log2 [σmax (Tm )] 2n 2n = log2 lim 1, Bnm−1 1 m = ≤ log2 lim Trace(T ) . C = lim m→∞ m→∞ m→∞ m 2n (20)
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The proof of (20) is similar to 1 1 1 1 1 1 1 1 1 B3 = 1 1 1 1 1 1 1 1 1
that of Lemma 2. As an example, B3 is calculated to be 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0
1 1 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0
1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0
1 1 1 1 0 1 1 0 1 0 1 1 1 1 1 1 0 0
1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0
1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 1 1 0 1 0 0 1 1 1 0 1 0 0
1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0
1 1 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0
and the corresponding upper bound is computed to be 0.4813. V. C OMPARISON
TO
1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0
1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 0 0 0
1 1 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 0
.
(21)
S OME E XISTING R ESULTS
In [1], Weeks et al. considered a similar problem on an integer lattice, which is equivalent to the capacity of the shaded part of Gm,n (Fig. 4) after being mapped to a hexagonal lattice. Similar to (10), the capacity in [1] is computed to be lim [log2 σmax (Xn )]/n. The following result shows that the capacity computed in this paper and n→∞ the one studied by Weeks are the same. Theorem 1. The following two limits exist and are equal log2 σmax (Xn ) log2 σmax (Tm ) = lim . n→∞ m→∞ n m Proof. Because of the existence of
n−1 2f (m, n) = lim [log2 1, Tm 1 ]/mn = lim [log2 cm,n 1, Xnm−hn −1 1 ]/mn lim
m,n→∞
m,n→∞
and (12), (16), the following iterated limits exist and are both equal to the double limits in (23)
n−1 1 log2 1, Tm log2 σmax (Tm ) lim = lim lim m→∞ m→∞ n→∞ m mn
log2 cm,n 1, Xnm−hn −1 1 log2 σmax (Xn ) lim = lim lim . n→∞ n→∞ m→∞ n mn
(22)
(23)
(24) (25)
They proposed a recursive method to compute Xn for n = 1, 2, · · · , and used log2 σmax (Xn ) n log2 σmax (Xn ) ≤C≤ (26) n+1 n n to bound the capacity. However, the convergence becomes slow since the size of the transfer matrix increases as fast as the Fibonacci sequence with n. As the size of the matrix increases, calculating its largest eigenvalue becomes computationally demanding. On the other hand, the approximation is not accurate for small n. Applying X18 to (26) gives the bound 0.4640 ≤ C ≤ 0.4897 (which requires the eigenvalues of a matrix of size F19 = 6765), while our calculated bound (which also involves a matrix of size F19 ), 0.4807676144 ≤ C ≤ 0.4813, is much tighter. Fig. 6 shows the accuracy of the bounds vs. computational complexity, which is in terms of the size of the largest matrix whose eigenvalues need to be calculated, for both the method used by Weeks and the one derived in this paper.
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Fig. 6: A comparison between the two methods: upper and lower bounds on the capacity vs. complexity (the middle two lines are the bounds obtained by our method) Weeks et al. [1] also obtained 0.482644 as an estimate of the capacity using Richardson extrapolation and conjectured that it is correct up to the sixth decimal digit. Unfortunately, that conjecture is not true as this estimate is greater than our rigorous upper bound 0.4813. Halevy et al. [7], [16] presented a different estimation which up to nine decimal is 0.480767622. That approximation is consistent with our lower and upper bounds. Besides this, the value given by Halevy et al. and our lower bound (0.480767614) are the same up to 7 decimals. VI. C ONCLUSION A new method was proposed to derive bounds on the capacity of constrained two-dimensional code on the hexagonal lattice. The methods rely on the zigzag patterns that we chose on the hexagonal lattice in order to obtain nonnegative and irreducible transfer matrices. R EFERENCES [1] W. Weeks and R. E. Blahut, “The capacity and coding gain of certain checkerboard codes,” Information Theory, IEEE Transactions on, vol. 44, no. 3, pp. 1193–1203, 1998. [2] Z. Nagy and K. Zeger, “Asymptotic capacity of two-dimensional channels with checkerboard constraints,” Information Theory, IEEE Transactions on, vol. 49, no. 9, pp. 2115–2125, 2003. [3] A. Gallopoulos, C. Heegard, and P. H. Siegel, “The power spectrum of run-length-limited codes,” Communications, IEEE Transactions on, vol. 37, no. 9, pp. 906–917, 1989. [4] F. Topsoe, “Some inequalities for information divergence and related measures of discrimination,” Information Theory, IEEE Transactions on, vol. 46, no. 4, pp. 1602–1609, 2000. [5] S. Forchhammer and J. Justesen, “Bounds on the capacity of constrained two-dimensional codes,” Information Theory, IEEE Transactions on, vol. 46, no. 7, pp. 2659–2666, 2000. [6] R. M. Roth, P. H. Siegel, and J. K. Wolf, “Efficient coding schemes for the hard-square model,” Information Theory, IEEE Transactions on, vol. 47, no. 3, pp. 1166–1176, 2001. [7] S. Halevy, J. Chen, R. M. Roth, P. H. Siegel, and J. K. Wolf, “Improved bit-stuffing bounds on two-dimensional constraints,” Information Theory, IEEE Transactions on, vol. 50, no. 5, pp. 824–838, 2004. [8] D. Brady and D. Psaltis, “Control of volume holograms,” JOSA A, vol. 9, no. 7, pp. 1167–1182, 1992. [9] D. Psaltis, M. A. Neifeld, A. Yamamura, and S. Kobayashi, “Optical memory disks in optical information processing,” Applied Optics, vol. 29, no. 14, pp. 2038–2057, 1990. ¨ [10] L. Fejes, “Uber die dichteste kugellagerung,” Mathematische Zeitschrift, vol. 48, no. 1, pp. 676–684, 1942. [11] J. Justesen and Y. M. Shtar’kov, “Combinatorial entropy of images,” Problemy Peredachi Informatsii, vol. 33, no. 1, pp. 3–11, 1997. [12] S. Forchhammer and J. Justesen, “Entropy bounds for constrained two-dimensional random fields,” Information Theory, IEEE Transactions on, vol. 45, no. 1, pp. 118–127, 1999. [13] H. S. Wilf, “The problem of the kings,” Electron. J. Combin, vol. 2, p. R3, 1995. [14] N. J. Calkin and H. S. Wilf, “The number of independent sets in a grid graph,” SIAM Journal on Discrete Mathematics, vol. 11, no. 1, pp. 54–60, 1998. [15] D. Lind and B. Marcus, An introduction to symbolic dynamics and coding. Cambridge University Press, 1995. [16] R. J. Baxter, “Hard hexagons: exact solution,” Journal of Physics A: Mathematical and General, vol. 13, no. 3, p. L61, 1980.
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Zhun Deng is a Ph.D. candidate in the School of Engineering and Applied Sciences, Harvard University. His current research are in statistical signal processing, information theory and fuzzy mathematics.
Jie Ding is a Ph.D. candidate in the School of Engineering and Applied Sciences, Harvard University. His current research are in cyclic difference sets, time series, information theory.
Mohammad Noshad is a postdoctoral fellow in the Electrical Engineering Department at Harvard University. He received his PhD in Electrical Engineering from the University of Virginia. He is a recipient of the “Best Paper Award” at IEEE Globecom 2012. His research interests include information and coding theory, statistical machine learning, free-space optical communications, and visible light communications.
Vahid Tarokh is a professor of applied mathematics in the School of Engineering and Applied Sciences, Harvard University. His current research interests are in data analysis, network security, optical surveillance, and radar theory.
Capacity of Hexagonal Checkerboard Codes Zhun Deng, Jie Ding, Mohammad Noshad, and Vahid Tarokh
Abstract In this paper, we propose a new method to bound the capacity of checkerboard codes on the hexagonal lattice. This produces rigorous bounds that are tighter than those commonly known.
arXiv:1506.02163v1 [cs.IT] 6 Jun 2015
Index Terms Capacity, constrained codes, hexagonal lattice, transfer matrix.
I. I NTRODUCTION HECKERBOARD codes are two-dimensional binary codes that are designed to satisfy specific constraints [1], [2]. An example is the two-dimensional rectangular binary arrays that satisfy the (d, k) run-length constraint— there are at most k 0’s and the number of 0’s between any neighboring 1’s is at least d in each row and column [3]–[7]. It may naturally arise from data storage on a surface [8], [9]. An example of the two dimensional (1, ∞) run-length constraint is shown in Fig. 1. In practice, the distance between two data recorded points on a recording device should be no less than a given threshold due to the physical and fidelity constraints. It is of interest to study the checkerboard code on the hexagonal lattice, because it is known that the lattice arrangement of circles with the highest density in two dimensional Euclidean space is the hexagonal packing arrangement, in which the centers of the circles are arranged on the hexagonal lattice [10]. Specifically, we study the capacity of the checkerboard code shown in Fig. 2, where only 0’s can be arranged in the six neighbors of any 1, while both 0 and 1 can be arranged in the six neighbors of any 0 (which is referred to as the hexagonal constraint). ∞ S S i = A2 . Let S1 ⊂ S2 ⊂ · · · be a sequence of finite subsets of two dimensional hexagonal lattice A2 such that
C
i=1
Let f (Si ) (i ∈ N) be the total number of distinct arrangements of 0’s and 1’s satisfying the hexagonal constraint given a two-dimensional region Si , and |Si | denotes the number of points inside Si . The capacity of the hexagonal checkerboard code is defined as log2 f (Si ) , (1) C ∗ = lim i→∞ |Si | if the limit exists. The subsets Si , i = 1, 2, · · · , considered in this work will mainly be the zigzag regions shown in Fig. 3. The existence of the capacity has been established in some prior papers [11], [12] using the subadditivity property of a double sequence. Weeks et al. [1] studied a similar problem on the integer lattice. They provided rigorous lower and upper bounds on the capacity and produced high precision approximations of using Richardson extrapolation. A detailed discussion on their results is included in Section V. Wilf [13] proposed the transfer matrix method to study the problem of arranging kings in a checkerboard. Calkin et al. [14] further used the transfer matrix to compute the bounds on the capacity of the diamond constraint shown in Fig. 1. However, their method cannot be directly used in calculating the capacity of hexagonal lattice. Inspired by Calkin et al.’s work, we propose a modified transfer matrix method that provides rigorous bounds on the capacity of the hexagonal constraint. The bounds presented in this work are tighter than those calculated by Weeks et al. [1], under the same computational complexity. II. T RANSFER M ATRIX
FOR
H EXAGONAL L ATTICE A RRAYS
A. Notations Let {Fi }∞ i=0 denote the Fibonacci numbers, i.e., F0 = 1, F1 = 1, and Fi = Fi−1 + Fi−2 . For any real-valued matrix A, let σmax (A), AT , and Trace(A) respectively denote the largest absolute value of the eigenvalues, the transpose, and the sum of all the diagonal elements of A. We represent the inner product of any two real-valued vectors x and y of the same size by hx, yi. The notations 1s×r and 0s×r respectively denote the s × r matrices of Z. Deng, J. Ding, M. Noshad and V. Tarokh are with the School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA. E-mail: [email protected] , [email protected] , [email protected] , [email protected] .
2
Fig. 1: Illustration of the diamond and square constraints
Fig. 2: Illustration of the hexagonal constraint all ones and all zeros. We drop the subscripts s × r when there is no ambiguity. We note that for any real-valued symmetric r × r matrix A and positive integer n, the following inequality holds (σmax (A))n ≥
h1r×1 , An 1r×1 i . h1r×1 , 1r×1 i
(2)
We provide the following lemma for the future convenience. It follows immediately from the Perron-Frobenius Theorem and Proposition 4.2.1 in [15]. Lemma 1. Let A be a real-valued r×r matrix that is nonnegative and irreducible. Then A has a positive eigenvector T v = [v[1], v[2], · · · , v[r]] with corresponding eigenvalue σmax (A). Besides this, the following inequality holds for any positive integer n rd e (σmax (A))n ≤ h1, An 1i ≤ (σmax (A))n (3) d e where e = min{v[1], v[2], · · · , v[r]} and d = max{v[1], v[2], · · · , v[r]}. B. Transfer Matrix Let Gm,n denote the zigzag shaped array on the hexagonal lattice shown in Fig. 3, where m is the number of points in each row and n is the number of rows. The row vectors are categorized into two different types: the ones whose first point lies on the left of its upper row (or lower row) are termed as “L” (left) type, while the others belong to “R” (right) type. Assume that the lowest row of Gm,n is of “R” type. We emphasize that the vector “100001“ of “L” type is different from “100001” of “R” type. Clearly, an “L” type vector cannot be attached to an “L” type vector, while an “R” type vector can be attached to an “L” type vector–either upwards or downwards (see Fig. 3).
Fig. 3: Illustration of the zigzag hexagonal array with m rows and n columns
3
Let f (m, n) denote the total number of valid arrangements of 0’s and 1’s in Gm,n satisfying the hexagonal constraint. The value of (1) for the given set of of regions therefore becomes C=
lim
m,n→∞
log2 f (m, n) . mn
(4)
We will provide lower and upper bounds on C in the rest part of the paper. A row vector is called “valid” if none of its two neighboring elements are both ones. Let am denote the number of valid row vectors of length m. Then am = Fm+1 . In fact, if a valid row starts with element zero, the remaining m − 1 elements have am−1 valid arrangements; if the row starts with one, the second element must be zero and the remaining m − 2 elements have am−2 valid arrangements. So, we obtain am = am−1 + am−2 . Besides, it is easy m to see that a1 = 2, a2 = 3, which results in am = Fm+1 . Let Am = {ui }2a i=1 represent the collection of all the “L” type and “R” type valid row vectors of length m (without loss of generality, the “R” type are arranged before the m belong to “R” type). Clearly, each row of Gm,n belongs to Am . “L” type within Am , i.e., {ui }ai=1 Next, we define the transfer matrix Tm to be a 2am × 2am binary matrix whose (i, j)th entry is 1 if uj can be attached to ui (as is depicted in Fig. 3) Tm [i, j] = , 1 ≤ i, j ≤ 2am . (5) 0 otherwise Note that Tm is a symmetric matrix due to its construction. Let f (m, n, i) denote the number of valid arrangements in Gm,n whose top row is ui . Then we obtain f (m, n + 1, j) =
2a m X
f (m, n, i)Tm [i, j].
(6)
i=1
Let fm,n = [f (m, n, 1), f (m, n, 2), · · · , f (m, n, 2am)]. Therefore, (6) can be rewritten in the following form fm,n+1 = Tm fm,n , and fm,1 = [1, 1 · · · , 1, 0, 0, · · · , 0]T
(am 1’s) ,
from which we get
n−1 f (m, n) = 1, Tm fm,1 .
(7)
Due to symmetry, f (m, n) remains the same if the lowest row of Gm,n is “L” type instead of “R” type, i.e.,
n−1 f (m, n) = 1, Tm (1 − fm,1 ) ,
(8)
Combining (7) and (8) yields
n−1 2f (m, n) = 1, Tm 1 ,
which leads to the following lemma.
(9)
Lemma 2. For any given positive integer m, we obtain i h h
1 i 1 n−1 1, Tm 1 n lim log2 f (m, n) n lim log2 log2 f (m, n) log2 [σmax (Tm )] lim = n→∞ = n→∞ = . (10) n→∞ nm m m m Proof. Tm is an irreducible matrix since the directed graph it represents is strongly-connected. In fact, we can build a path from one vertex to any other vertex by inserting vertices that represent the all-zero vectors. Furthermore, Lemma 1 implies that there are positive constants d and e that depend on m such that
2am d e n−1 (σmax (Tm ))n−1 ≤ 1, Tm 1 ≤ (σmax (Tm ))n−1 . (11) d e Raising both sides of (11) to the power of 1/(n − 1) and letting n → ∞, we get
1/(n−1) n−1 lim 1, Tm 1 = σmax (Tm ) (12) n→∞
which implies the existence of the limit in (10). From Lemma 2 and since
lim [log2 f (m, n)]/mn exists, the iterated limit lim lim [log2 f (m, n)]/mn also
m,n→∞
m→∞ n→∞
exists and they are equal. Therefore, we only need to compute bounds on the iterated limit.
4
Example 1. If m = 3, the valid rows are [0, 0, 0]R , [0, 0, 1]R , [0, 1, 0]R , [1, 0, 0]R , [1, 0, 1]R , [0, 0, 0]L , [0, 0, 1]L , [0, 1, 0]L , [1, 0, 0]L , [1, 0, 1]L , and the transfer matrix is 1 1 1 1 1 1 0 1 1 0 0 A T3 = , where A = 1 0 0 1 0 . AT 0 1 1 0 0 0 1 0 0 0 0 III. L OWER B OUNDS
ON
T HE C APACITY
We denote the shaded part of Gm,n that is formed by concatenating n column vectors by Gshaded m,n , as shown in n . Fig. 4. Similar to the arguments in Subsection II-B, there are an = Fn+1 valid column vectors, denoted by {wi }ai=1 We define Xn to be a an × an binary matrix whose (i, j)th entry is 1 if wj can be attached to wi , 1 ≤ i, j ≤ an . (13) Xn [i, j] = 0 otherwise Example 2. If n = 3, the valid columns are [0, 0, 0] , [0, 0, 1] , [0, 1, 0] , [1, 0, 0] , [1, 0, 1], and the transfer matrix is 1 1 1 1 1 1 0 0 1 0 X3 = 1 1 0 0 0 . 1 1 1 0 0 1 0 0 0 0
One can observe that each row of of Gshaded m,n is of length m − hn , where hn = [n/2] ([u] denotes the largest integer that is no larger than u). Similar to the derivation of (7), the
number of distinct arrangements satisfying the m−hn −1 hexagonal constraint in Gshaded is calculated to be 1, X 1 . Besides this, the number of points in Gm,n but m,n n is nm − n(m − h ) = nh . Thus, there exists a positive integer cm,n , which depends on m, n and not in Gshaded n n m,n upper-bounded by 2nhn , such that 1
1
n−1 1, Tm 1 = f (m, n) = cm,n 1, Xnm−hn −1 1 . (14) 2 2 For example, when n = 5 (Fig. 4), we have h5 = 2 and 1 ≤ c5 ≤ 210 .
Fig. 4: Illustration of the shaded part of Gm,n Because of (14) and the fact that Tm is symmetric, for any positive integer q we obtain m−h
(σmax (Tm ))n ≥ ≥
−1
n+2q+1 n+2q q n q 1i h1, Tm 1i cm,n+2q+1 h1, Xn+2q+1 hTm 1, Tm Tm 1i = = q q 2q m−h2q+1 −1 hTm 1, Tm 1i c m,2q+1 h1, Tm 1i h1, X2q+1 1i
1 2(2q+1)h2q+1
m−h
n+2q+1 h1, Xn+2q+1
−1
1i
m−h −1 h1, X2q+1 2q+1 1i
.
As a specific case, we choose n = 1, q = 8 and obtain σmax (Tm ) ≥
1 217h17
m−h18 −1 h1, X18 1i . m−h17 −1 h1, X17 1i
(15)
5
n 1 2 3 4 5 6
λmax 1.618 2.147 3.054 4.233 5.922 8.256
n 7 8 9 10 11 12
λmax 11.526 16.082 22.443 31.319 43.706 60.991
n 13 14 15 16 17 18
λmax 85.112 118.773 165.747 231.297 322.773 450.425
TABLE I: The largest eigenvalue of Xn
Fig. 5: Illustration of Bn Following a similar argument as in the proof of Lemma 2, Xn is irreducible and the following equality holds for any fixed n > 0
1 1/m σmax (Xn ) = lim h1, Xnm 1i = lim cm,n 1, Xnm−hn −1 1 m . (16) m→∞
m→∞
We thus obtain
log2 [σmax (Tm )] ≥ log2 lim C = lim m→∞ m→∞ m
1 217h17
m−h18 −1 h1, X18 1i m−h17 −1 h1, X17 1i
! m1
= log2
σmax (X18 ) σmax (X17 )
= 0.4807676144.
The largest eigenvalues of the transfer matrix Xn are listed in Table I for n = 1, 2, · · · , 18 . IV. U PPER B OUNDS
ON
T HE C APACITY
Since Tm is a real and symmetric matrix, the following inequality holds 1 σmax (Tm ) ≤ Trace(Tm )2n ) 2n ,
where
2n Trace(Tm )=
X
Tm [x0 , x1 ] Tm [x1 , x2 ] · · · Tm [x2n−1 , x0 ].
(17) (18)
1≤x0 ,x1 ,··· ,x2n−1 ≤2am
Clearly, the right hand side of (18) is the total number of valid arrangements of the row vectors (ux0 , ux1 , · · · , ux2n−1 , ux0 ) in a zigzag way from bottom to top, as shown in Fig. 5. We calculate the sum in (18) by considering the zigzag array as a concatenation of (2n + 1) × 1 valid column vectors (Fig. 5). The top element should be the same as the bottom one in each column, so that the first row and the last row become the same. We denote the n . The number of elements in Bn , denoted by bn , is F2n−2 + F2n collection of valid column vectors as Bn = {vi }bi=1 (n > 1) due to the following reason. If the bottom and top elements of a (2n + 1) × 1 column vector are 0, the remaining 2n − 1 elements have F2n valid arrangements; if the bottom and top elements are 1, the elements next to them must be 0, and the remaining 2n − 3 elements have F2n−2 valid arrangements. We define Bn to be a bn × bn binary matrix whose (i, j)th entry is 1 if vj can be attached to vi Bn [i, j] = , 1 ≤ i, j ≤ bn . (19) 0 otherwise
2n Using an equation similar to (6), we get Trace(Tm ) = 1, Bnm−1 1 , which leads to 1 h m1
1 i 2n 1 log2 [σmax (Bn )] log2 [σmax (Tm )] 2n 2n = log2 lim 1, Bnm−1 1 m = ≤ log2 lim Trace(T ) . C = lim m→∞ m→∞ m→∞ m 2n (20)
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The proof of (20) is similar to 1 1 1 1 1 1 1 1 1 B3 = 1 1 1 1 1 1 1 1 1
that of Lemma 2. As an example, B3 is calculated to be 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0
1 1 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0
1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0
1 1 1 1 0 1 1 0 1 0 1 1 1 1 1 1 0 0
1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0
1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 1 1 0 1 0 0 1 1 1 0 1 0 0
1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0
1 1 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0
and the corresponding upper bound is computed to be 0.4813. V. C OMPARISON
TO
1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0
1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 0 0 0
1 1 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 0
.
(21)
S OME E XISTING R ESULTS
In [1], Weeks et al. considered a similar problem on an integer lattice, which is equivalent to the capacity of the shaded part of Gm,n (Fig. 4) after being mapped to a hexagonal lattice. Similar to (10), the capacity in [1] is computed to be lim [log2 σmax (Xn )]/n. The following result shows that the capacity computed in this paper and n→∞ the one studied by Weeks are the same. Theorem 1. The following two limits exist and are equal log2 σmax (Xn ) log2 σmax (Tm ) = lim . n→∞ m→∞ n m Proof. Because of the existence of
n−1 2f (m, n) = lim [log2 1, Tm 1 ]/mn = lim [log2 cm,n 1, Xnm−hn −1 1 ]/mn lim
m,n→∞
m,n→∞
and (12), (16), the following iterated limits exist and are both equal to the double limits in (23)
n−1 1 log2 1, Tm log2 σmax (Tm ) lim = lim lim m→∞ m→∞ n→∞ m mn
log2 cm,n 1, Xnm−hn −1 1 log2 σmax (Xn ) lim = lim lim . n→∞ n→∞ m→∞ n mn
(22)
(23)
(24) (25)
They proposed a recursive method to compute Xn for n = 1, 2, · · · , and used log2 σmax (Xn ) n log2 σmax (Xn ) ≤C≤ (26) n+1 n n to bound the capacity. However, the convergence becomes slow since the size of the transfer matrix increases as fast as the Fibonacci sequence with n. As the size of the matrix increases, calculating its largest eigenvalue becomes computationally demanding. On the other hand, the approximation is not accurate for small n. Applying X18 to (26) gives the bound 0.4640 ≤ C ≤ 0.4897 (which requires the eigenvalues of a matrix of size F19 = 6765), while our calculated bound (which also involves a matrix of size F19 ), 0.4807676144 ≤ C ≤ 0.4813, is much tighter. Fig. 6 shows the accuracy of the bounds vs. computational complexity, which is in terms of the size of the largest matrix whose eigenvalues need to be calculated, for both the method used by Weeks and the one derived in this paper.
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Fig. 6: A comparison between the two methods: upper and lower bounds on the capacity vs. complexity (the middle two lines are the bounds obtained by our method) Weeks et al. [1] also obtained 0.482644 as an estimate of the capacity using Richardson extrapolation and conjectured that it is correct up to the sixth decimal digit. Unfortunately, that conjecture is not true as this estimate is greater than our rigorous upper bound 0.4813. Halevy et al. [7], [16] presented a different estimation which up to nine decimal is 0.480767622. That approximation is consistent with our lower and upper bounds. Besides this, the value given by Halevy et al. and our lower bound (0.480767614) are the same up to 7 decimals. VI. C ONCLUSION A new method was proposed to derive bounds on the capacity of constrained two-dimensional code on the hexagonal lattice. The methods rely on the zigzag patterns that we chose on the hexagonal lattice in order to obtain nonnegative and irreducible transfer matrices. R EFERENCES [1] W. Weeks and R. E. Blahut, “The capacity and coding gain of certain checkerboard codes,” Information Theory, IEEE Transactions on, vol. 44, no. 3, pp. 1193–1203, 1998. [2] Z. Nagy and K. Zeger, “Asymptotic capacity of two-dimensional channels with checkerboard constraints,” Information Theory, IEEE Transactions on, vol. 49, no. 9, pp. 2115–2125, 2003. [3] A. Gallopoulos, C. Heegard, and P. H. Siegel, “The power spectrum of run-length-limited codes,” Communications, IEEE Transactions on, vol. 37, no. 9, pp. 906–917, 1989. [4] F. Topsoe, “Some inequalities for information divergence and related measures of discrimination,” Information Theory, IEEE Transactions on, vol. 46, no. 4, pp. 1602–1609, 2000. [5] S. Forchhammer and J. Justesen, “Bounds on the capacity of constrained two-dimensional codes,” Information Theory, IEEE Transactions on, vol. 46, no. 7, pp. 2659–2666, 2000. [6] R. M. Roth, P. H. Siegel, and J. K. Wolf, “Efficient coding schemes for the hard-square model,” Information Theory, IEEE Transactions on, vol. 47, no. 3, pp. 1166–1176, 2001. [7] S. Halevy, J. Chen, R. M. Roth, P. H. Siegel, and J. K. Wolf, “Improved bit-stuffing bounds on two-dimensional constraints,” Information Theory, IEEE Transactions on, vol. 50, no. 5, pp. 824–838, 2004. [8] D. Brady and D. Psaltis, “Control of volume holograms,” JOSA A, vol. 9, no. 7, pp. 1167–1182, 1992. [9] D. Psaltis, M. A. Neifeld, A. Yamamura, and S. Kobayashi, “Optical memory disks in optical information processing,” Applied Optics, vol. 29, no. 14, pp. 2038–2057, 1990. ¨ [10] L. Fejes, “Uber die dichteste kugellagerung,” Mathematische Zeitschrift, vol. 48, no. 1, pp. 676–684, 1942. [11] J. Justesen and Y. M. Shtar’kov, “Combinatorial entropy of images,” Problemy Peredachi Informatsii, vol. 33, no. 1, pp. 3–11, 1997. [12] S. Forchhammer and J. Justesen, “Entropy bounds for constrained two-dimensional random fields,” Information Theory, IEEE Transactions on, vol. 45, no. 1, pp. 118–127, 1999. [13] H. S. Wilf, “The problem of the kings,” Electron. J. Combin, vol. 2, p. R3, 1995. [14] N. J. Calkin and H. S. Wilf, “The number of independent sets in a grid graph,” SIAM Journal on Discrete Mathematics, vol. 11, no. 1, pp. 54–60, 1998. [15] D. Lind and B. Marcus, An introduction to symbolic dynamics and coding. Cambridge University Press, 1995. [16] R. J. Baxter, “Hard hexagons: exact solution,” Journal of Physics A: Mathematical and General, vol. 13, no. 3, p. L61, 1980.
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Zhun Deng is a Ph.D. candidate in the School of Engineering and Applied Sciences, Harvard University. His current research are in statistical signal processing, information theory and fuzzy mathematics.
Jie Ding is a Ph.D. candidate in the School of Engineering and Applied Sciences, Harvard University. His current research are in cyclic difference sets, time series, information theory.
Mohammad Noshad is a postdoctoral fellow in the Electrical Engineering Department at Harvard University. He received his PhD in Electrical Engineering from the University of Virginia. He is a recipient of the “Best Paper Award” at IEEE Globecom 2012. His research interests include information and coding theory, statistical machine learning, free-space optical communications, and visible light communications.
Vahid Tarokh is a professor of applied mathematics in the School of Engineering and Applied Sciences, Harvard University. His current research interests are in data analysis, network security, optical surveillance, and radar theory.
