A. Kato and K. Zeger, On the capacity of two-dimensional run length constrained channels. IEEE Trans. Inform. Theory 45 (1999), 1527-1540.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... A is a shift, the entropy of X is de ned as h(X) lim ; 1) where Bn (X) is the set of square blocks of X of size n. We will always use the base 2 for logarithms. The existence of the limit in (1) is proved for instance in [12, Proposition 4.1. 8] for the one dimensional case and in [9] for the multi dimensional one. For each 2 Z , the set Dn provides, by translation, a neighborhood of , that is the set D( n) Dn = n; n] Given a subset E Z for each k 2 N we denote by k : 2E D( k) and E : f 2 E j D( k) Eg the k closure of E ....

A. Kato and K. Zeger, On the capacity of two-dimensional run-length constrained channels, IEEE Trans. Inform. Theory, 45 (1999), pp. 1527{ 1540.

.... is equal to h(SH;V (m) lim p 1 log Per p (SH;V (m) log m where Per p (SH;V (m) is the number of periodic points of period p of SH;V (m) see [14] Since SH;V (m) is a system of nite type we have that Per p (SH;V (m) trace(T m ) The following result is already known, see [11, 13, 18]. Corollary 4 The entropy of the system is given by the limit = lim m 1 (log m ) m where m is the dominant eigenvalue of ( a 0 ) Proof. Let P = PH;V for short. Words m n can be cut by Euclidean division in words of shorter length. We pose n = r q with 0 r q and q a ....

A. Kato and K. Zeger, On the capacity of two-dimensional run length constrained channels. IEEE Trans. Inform. Theory 45 (1999), 1527-1540.

....examined two dimensional constrained codes. Recent work in twodimensional runlength limited codes, which arise in magnetic recording applications, includes multitrack recording [17] 16] work in cascading arrays [4] and calculation of capacity of two and three dimensional RLL codes [11] 15] [10]. The two dimensional codes examined in this paper are related to two dimensional RLL codes. Work related to computing capacity for general constraints for two dimensional codes also exists [3] 2] 19] 6] 5] Finally, work in constructing e#cient encoders and decoders for constrained ....

A. Kato and K. Zeger. On the capacity of two-dimensional run-length constrained channels. IEEE Transactions on Information Theory, 45(5):1527--40, July 1999.

....path in H. Letting S[ m] stand for the set of Theta m arrays in S, the capacity of a two dimensional constraint is defined by ;m 1 (1= m) Delta log 2 jS[ m]j : 1) As in the one dimensional case, the limit indeed exists by sub additivity (see Burton and Steif [2] and Kato and Zeger [12]; the result in [12] is stated for the special case of runlength constraints, but the proof actually applies to all two dimensional constraints) In Section 5, we apply parallel encoding to show that for two dimensional constraints that satisfy certain properties, capacity can be approached by ....

.... m] stand for the set of Theta m arrays in S, the capacity of a two dimensional constraint is defined by ;m 1 (1= m) Delta log 2 jS[ m]j : 1) As in the one dimensional case, the limit indeed exists by sub additivity (see Burton and Steif [2] and Kato and Zeger [12] the result in [12] is stated for the special case of runlength constraints, but the proof actually applies to all two dimensional constraints) In Section 5, we apply parallel encoding to show that for two dimensional constraints that satisfy certain properties, capacity can be approached by fixed rate block ....

A. Kato, K. Zeger, On the capacity of two-dimensional run length constrained channels, IEEE Trans. Inform. Theory, 45 (1999), 1527--1540.

....r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r 1 2 n Gamma1 0 1 2 Gamma(m Gamma1) 1 2 m Gamma1 Figure 1: Parallelogram Delta m;n . The limits indeed exist and are equal [5] [16]. The value of cap(S) is known to be approximately 0:5878911162; see [6] 10] 11] 29] Much less is known about efficient (i.e. polynomial time, or low complexity) high rate coding schemes for this constraint. In [26] the idea of two dimensional bit stuffing was introduced, resulting in a ....

A. Kato, K. Zeger, On the capacity of two-dimensional run length constrained channels, IEEE Trans. Inform. Theory, 45 (1999), 1527--1540.

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A. Kato and K. Zeger, "On the Capacity of Two-Dimensional Run Length Constrained Channels," IEEE Transactions on Information Theory, vol. 45, no. 5, July 1999, pp. 1527--1540.

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A. Kato and K. Zeger. On the capacity of two-dimensional run length constrained channels. IEEE Trans. Inform. Theory, 45(4):1527--1540, July 1999.

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A. Kato and K. Zeger. On the capacity of two-dimensional run length constrained channels. IEEE Trans. Inform. Theory, 45(4):1527--1540, July 1999.

....is denoted by N m ;m ;111;m and the corresponding capacity is defined as d;k =lim m ;m ;111m 1 m ;m ;111m By exchanging the roles of 0 and 1 it can be seen that C 0;1 = C 1;1 for all n 1. A simple proof of the existence of the two dimensional (d; k) capacities can be found in [1], and the proof can be generalized to n dimensions. It is known (e.g. see [2] that the one dimensional (0; 1) constrained capacity is the logarithm of the golden ratio, i.e. 1) 0;1 = log 2 1 p 5 =0:694242 111 and in [3] very close upper and lower bounds were given for the ....

....positions 0;1 0:587891161868: 1) These bounds were also independently obtained to eight decimal positions in [5] A lower bound of C 0;1 0:5831 was obtained in [6] by using an implementable encoding procedure known as bit stuffing. The known bounds on C 0;1 have played a useful role in [1] for obtaining bounds on other (d; k) constraints in two dimensions. The three dimensional (0; 1) constrained bounds given in this correspondence can play a similar role for obtaining different three dimensional bounds, and are also of theoretical interest. In fact, a recent tutorial paper [7] ....

A. Kato and K. Zeger, "On the capacity of two-dimensional run length constrained channels," IEEE Trans. Inform. Theory, vol. 45, pp. 1527--1540, July 1999.

....the n dimensional (d; k) constraint (or of the constrained channel) is m 1 log 2 j (R m )j where R m = f0; 1; m 1g (there are various other equivalent definitions) The exact value of the capacity is not known in general. If d = k then C = 0, and it has been shown [10] [11] that if k d 1 and n 2 then C = 0 ( k = d 1. Numerical upper and lower bounds on C 1;1 were established in [3] and these bounds were later improved in [23] and then in [13] The best known bounds on C 1;1 agree in the first 9 decimal places as 1;1 0:587891161868: Numerical ....

A. Kato and K. Zeger. On the capacity of two-dimensional run length constrained channels. IEEE Trans. Inform. Theory, 45(4):1527--1540, July 1999.

....In this paper we focus on the asymptotic behavior of the capacity of two dimensional channels satisfying checkerboard constraints. In the special case of the two dimensional (d; 1) run length constrained channel, the asymptotic behavior of the capacity is well understood. It was shown in [16] that the capacity decays to zero at the exact rate (log 2 d) d as d 1. For a general checkerboard constraint, the asymptotics analogous to run length constraints are when the constraint S retains its shape but is inflated in size in the form S as 1. As goes to infinity, the amount of ....

....can be extended to an S valid labeling of R by making the labeling equal 0 outside of the subset. The number of S valid labelings of a set V R is denoted by NS (V ) The capacity CS corresponding to the checkerboard constraint S is CS = lim log 2 NS : 3) A proof given in [16] shows that the above limit exists. An example of a checkerboard constraint is a run length constraint. For each non negative integer d, the twodimensional (d; 1) run length constraint is defined as the following subset of R S d;1 = f(0; x) d x dg [ f(x; 0) d x dg (4) The ....

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A. Kato and K. Zeger. On the capacity of two-dimensional run length constrained channels. IEEE Trans. Info. Theory, 45:1527--1540, July 1999.

....satisfies lim n 1 Cn Delta ff 2 n log 2 ff 2 n = 4 A(S) I. Introduction One dimensional run length constraints are important in magnetic recording applications and two dimensional run length constraints have recently gained interest due to optical recording applications [1, 2, 3]. A two dimensional run length constraint requires that a binary labeling of the integer lattice Z 2 have a specified minimum and maximum number of zeros between consecutive ones both horizontally and vertically. Additional constraints, such as run length constraints along diagonals can also be ....

....of valid labelings of the rectangle R (n;m) 0;0) with respect to V. The capacity CV corresponding to a set V ae Z 2 including the origin is defined as CV = lim m;n 1 log 2 NV (m Gamma 1; n Gamma 1) mn : This work was supported in part by the National Science Foundation. The proof in [1] can be generalized to show that the above limit exists. III. The asymptotic capacity of the square constraint In this section S ae R 2 will denote a square centered at the origin, whose sides are parallel to the coordinate axes. Let S = S Z 2 , and let ff n be a sequence of positive real ....

A. Kato and K. Zeger, "On the Capacity of Two-Dimensional Run Length Constrained Channels," IEEE Trans. Inform. Theory. vol. 45, July 1999, pp.1527--1540.

....C (n) d;k = lim m 1 ;m 2 ; m n 1 log 2 N (n;d;k) m 1 ;m 2 ; m n m1m2 Delta Delta Delta mn ; where N (n;d;k) m 1 ;m 2 ; m n denotes the number of (d; k) constrained patterns on an m1 Theta m2 Theta Delta Delta Delta Theta mn hyper rectangle. A simple proof was given in [5] that shows the existence of twodimensional (d; k) capacities, and a slight modification of the proof can show that the n dimensional (d; k) capacities exist. The capacity C (n) d;k represents the maximum number of bits of information that can be stored asymptotically per unit volume in ....

....extended to three dimensions in [7] In [9] an encoding procedure for the 2dimensional (d; 1) constraint was given for all positive integer d s, and in [8] an encoding procedure for the 2 dimensional (0; 1) constraint was given whose coding rating comes very close to the capacity. It was shown [5] that whenever k d 1, the 2 dimensional capacity is zero if and only if k = d 1. II. Main Results We present two main results that characterize the zero capacity region for finite dimensions and in the limit of large dimensions. The first result generalizes the zero capacity characterization ....

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A. Kato and K. Zeger, "On the Capacity of Two-Dimensional Run Length Constrained Channels," IEEE Trans. Inform. Theory. vol. 45, no. 4, July 1999, pp.1527--1540.

....the number of m Theta n rectangles that are (d1 ; k1 ; d2 ; k2) constrained. If d = d1 = d2 and k = k1 = k2 (this is called the symmetric constraint) then the two dimensional (d1 ; k1 ; d2 ; k2) capacity is called the two dimensional (d; k) capacity, and is denoted by C d;k . A proof was given in [3] that shows the two dimensional (d; k) capacities exist, and essentially the same proof shows that the Cd 1 ;k 1 ;d 2 ;k 2 exist. The two dimensional asymmetric positive capacity region is the set f(d1 ; k1 ; d2 ; k2 ) Cd 1 ;k 1 ;d 2 ;k 2 0g: A basic question is to determine which ....

....1 ;d 2 ;k 2 0g: A basic question is to determine which constraints actually lie in the positive capacity region and which do not. For the symmetric constraints, it was shown in [1] that C1;2 = 0 and a complete characterization of which (d; k) integer pairs yield positive capacities was given in [3] and is stated as the proposition below. Proposition 1 C d;k 0 if and only if k Gamma d 2 or (d; k) 0; 1) II. Main Results In the present paper we determine whether or not the twodimensional capacity is positive, for a large set of asymmetric constraints (d1 ; k1 ; d2 ; k2 ) and the ....

A. Kato and K. Zeger, "On the Capacity of Two-Dimensional Run Length Constrained Channels," IEEE Trans. Inform. Theory, vol. 45, no. 5, July 1999, pp. 1527--1540.

....The n dimensional (d; k) capacity is de ned as C (n) d;k = lim m1;m2 ; mn 1 log 2 N (n;d;k) m 1 ;m 2 ; m n m 1 m 2 m n ; where N (n;d;k) m1;m2 ; mn denotes the number of (d; k) constrained patterns on an m 1 m 2 m n hyper rectangle. A simple proof was given in [5] that shows the existence of two dimensional (d; k) capacities, and a slight modi cation of the proof can show that the n dimensional (d; k) capacities exist. The capacity C (n) d;k represents the maximum number of bits of information that can be stored asymptotically per unit volume in ....

....if k = d the capacity is zero, and if d = 0 the capacity is positive for all k 1. In one dimension the capacity is positive whenever k d 0. The capacity is known to be a monotonically nonincreasing function of n and d and a monotonically nondecreasing function of k. It was recently shown [5] that whenever k d 1, the 2 dimensional capacity is zero if and only if k = d 1. These facts are summarized in our Lemma 1. Some interesting facts are known about the capacities for d = 0 and k = 1 in three and lower dimensions. In one dimension, N (1;0;1) m is known [6] to be a Fibonacci ....

[Article contains additional citation context not shown here]

A. Kato and K. Zeger, \On the Capacity of Two-Dimensional Run Length Constrained Channels," IEEE Trans. Information Theory. vol. 45, no. 4, July 1999, pp.1527-1540.

....The n dimensional (d, k) capacity is defined as C (n) d,k = lim m 1 ,m 2 , m n## log 2 N (n;d,k) m 1 ,m 2 , m n m 1 m 2 m n , where N (n;d,k) m 1 ,m 2 , mn denotes the number of (d, k) constrained patterns on an m 1 m 2 m n hyper rectangle. A simple proof was given in [5] that shows the existence of two dimensional (d, k) capacities, and a slight modification of the proof can show that the n dimensional (d, k) capacities exist. The capacity C (n) d,k represents the maximum number of bits of information that can be stored asymptotically per unit volume in ....

....if k = d the capacity is zero, and if d = 0 the capacity is positive for all k # 1. In one dimension the capacity is positive whenever k d# 0. The capacity is known to be a monotonically nonincreasing function of n and d and a monotonically nondecreasing function of k. It was recently shown [5] that whenever k d# 1, the 2 dimensional capacity is zero if and only if k = d 1. These facts are summarized in our Lemma 1. Some interesting facts are known about the capacities for d =0andk = 1 in three and lower dimensions. In one dimension, N (1;0,1) m is known [6] to be a Fibonacci ....

[Article contains additional citation context not shown here]

A. Kato and K. Zeger, "On the Capacity of Two-Dimensional Run Length Constrained Channels," IEEE Trans. Information Theory. vol. 45, no. 4, July 1999, pp.1527--1540.

....If d = d 1 = d 2 and k = k 1 = k 2 (this is called the symmetric constraint) then the twodimensional (d 1 ; k 1 ; d 2 ; k 2 ) capacity is called the two dimensional (d; k) capacity, and is denoted by C d;k . Two dimensional run length constraints have recently become a focus of increased study [1, 2, 4, 5, 6, 7, 9, 15, 16, 21]. A proof was given in [9] that shows the twodimensional (d; k) capacities exist, and essentially the same proof shows that the C d1 ;k 1 ;d 2 ;k 2 exist. The two dimensional asymmetric positive capacity region is the set f(d 1 ; k 1 ; d 2 ; k 2 ) C d1 ;k 1 ;d 2 ;k 2 0g: It is of interest ....

....symmetric constraint) then the twodimensional (d 1 ; k 1 ; d 2 ; k 2 ) capacity is called the two dimensional (d; k) capacity, and is denoted by C d;k . Two dimensional run length constraints have recently become a focus of increased study [1, 2, 4, 5, 6, 7, 9, 15, 16, 21] A proof was given in [9] that shows the twodimensional (d; k) capacities exist, and essentially the same proof shows that the C d1 ;k 1 ;d 2 ;k 2 exist. The two dimensional asymmetric positive capacity region is the set f(d 1 ; k 1 ; d 2 ; k 2 ) C d1 ;k 1 ;d 2 ;k 2 0g: It is of interest to determine the exact ....

[Article contains additional citation context not shown here]

A. Kato and K. Zeger, "On the Capacity of Two-Dimensional Run Length Constrained Channels," IEEE Trans. Inform. Theory, vol. 45, no. 5, July 1999, pp. 1527--1540.

....is defined as C (n) d;k = lim m1;m2 ; mn 1 log 2 N (d;k) m1;m2 ; mn m 1 m 2 Delta Delta Delta m n : By exchanging the roles of 0 and 1 it can be seen that C (n) 0;1 = C (n) 1;1 for all n 1. A simple proof of the existence of the 2 dimensional (d; k) capacities can be found in [1], and the proof can be generalized to n dimensions. It is known (e.g. see [2] that the 1 dimensional (0; 1) constrained capacity is the logarithm of the golden ratio, i.e. C (1) 0;1 = log 2 1 p 5 2 = 0:694242 : and in [3] very close upper and lower bounds were given for the ....

....end for more details) now agreeing in 9 decimal positions: 0:587891161775 C (2) 0;1 0:587891161868 : 1) A lower bound of C (2) 0;1 0:5831 was obtained in [5] by using an implementable encoding procedure known as bit stuffing . The known bounds on C (2) 0;1 have played a useful role in [1] for obtaining bounds on other (d; k) constraints in two dimensions. The 3 dimensional (0; 1) constrained bounds given in the present paper can play a similar role for obtaining different 3 dimensional bounds, and are also of theoretical interest. In fact, a recent tutorial paper [6] discusses ....

A. Kato and K. Zeger, "On the Capacity of Two-Dimensional Run Length Constrained Channels," IEEE Trans. Info. Theory, vol. 45, pp. 1527--1540, July 1999.

....as C (n) d;k = lim m1;m2 ; mn 1 log 2 N (n;d;k) m1;m2 ; mn m 1 m 2 Delta Delta Delta m n ; where N (n;d;k) m1;m2 ; mn denotes the number of (d; k) constrained patterns on an m 1 Theta m 2 Theta Delta Delta Delta Theta m n hyper rectangle. A simple proof was given in [5] that shows the existence of two dimensional (d; k) capacities, and a slight modification of the proof can show that the n dimensional (d; k) capacities exist. The capacity C (n) d;k represents the maximum number of bits of information that can be stored asymptotically per unit volume in ....

....if k = d the capacity is zero, and if d = 0 the capacity is positive for all k 1. In one dimension the capacity is positive whenever k d 0. The capacity is known to be a monotonically nonincreasing function of n and d and a monotonically nondecreasing function of k. It was recently shown [5] that whenever k d 1, the 2 dimensional capacity is zero if and only if k = d 1. These facts are summarized in our Lemma 1. Some interesting facts are known about the capacities for d = 0 and k = 1 in three and lower dimensions. In one dimension, N (1;0;1) m is known [6] to be a Fibonacci ....

[Article contains additional citation context not shown here]

A. Kato and K. Zeger, "On the Capacity of Two-Dimensional Run Length Constrained Channels," IEEE Trans. Information Theory. vol. 45, no. 4, July 1999, pp.1527--1540.

....capacity is defined as C (n) d;k = lim m1;m2 ; mn 1 log 2 N (d;k) m1;m2 ; mn m 1 m 2 mn : By exchanging the roles of 0 and 1 it can be seen that C (n) 0;1 = C (n) 1;1 for all n 1. A simple proof of the existence of the 2 dimensional (d; k) capacities can be found in [1], and the proof can be generalized to n dimensions. It is known (e.g. see [2] that the 1 dimensional (0; 1) constrained capacity is the logarithm of the golden ratio, i.e. C (1) 0;1 = log 2 1 p 5 2 = 0:694242 : and in [3] very close upper and lower bounds were given for the ....

....for more details) now agreeing in 9 decimal positions: 0:587891161775 C (2) 0;1 0:587891161868 : 1) A lower bound of C (2) 0;1 0:5831 was obtained in [5] by using an implementable encoding procedure known as bit stuffing . The known bounds on C (2) 0;1 have played a useful role in [1] for obtaining bounds on other (d; k) constraints in two dimensions. The 3 dimensional (0; 1) constrained bounds given in the present paper can play a similar role for obtaining different 3 dimensional bounds, and are also of theoretical interest. In fact, a recent tutorial paper [6] discusses an ....

A. Kato and K. Zeger, "On the Capacity of Two-Dimensional Run Length Constrained Channels, " IEEE Trans. Info. Theory, 1999 (to appear).

....(d; k) capacity is defined as C (n) d;k = lim m 1 ;m 2 ; m n 1 log 2 N (n;d;k) m 1 ;m 2 ; mn m 1 m 2 m n ; where N (n;d;k) m 1 ;m 2 ; m n denotes the number of (d; k) constrained patterns on an m 1 m 2 m n hyper rectangle. A simple proof was given in [5] that shows the existence of two dimensional (d; k) capacities, and a slight modification of the proof can show that the n dimensional (d; k) capacities exist. The capacity C (n) d;k represents the maximum number of bits of information that can be stored asymptotically per unit volume in ....

....if k = d the capacity is zero, and if d = 0 the capacity is positive for all k 1. In one dimension the capacity is positive whenever k d 0. The capacity is known to be a monotonically nonincreasing function of n and d and a monotonically nondecreasing function of k. It was recently shown [5] that whenever k d 1, the 2 dimensional capacity is zero if and only if k = d 1. These facts are summarized in our Lemma 1. Some interesting facts are known about the capacities for d = 0 and k = 1 in three and lower dimensions. In one dimension, N (1;0;1) m is known [6] to be a ....

[Article contains additional citation context not shown here]

A. Kato and K. Zeger, "On the Capacity of Two-Dimensional Run Length Constrained Channels," IEEE Trans. Information Theory. vol. 45, no. 4, July 1999 (to appear).

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A. Kato and K. Zeger, On the capacity of two-dimensional run length constrained channels. IEEE Trans. Inform. Theory 45 (1999), 1527-1540.

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A. Kato and K. Zeger, "On the capacity of two-dimensional run length constrained channels", IEEE Trans. Info. Theory, vol. IT-45, pp. 1527--1540, 1999.

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A. Kato, K. Zeger, On the capacity of two-dimensional run-length constrained channels, IEEE Trans. Inform. Theory, 45 (1999), 1527-1540.

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A. Kato and K. Zeger, "On the capacity of two-dimensional run-length constrained channels," IEEE Trans. Inform. Theory, vol. 45, no. 5, pp. 1527--1540, July 1999.

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