Improved Upper Bounds for Codes With Unequal Error Protection
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 7, JULY 2006
[8] D. Joyner and A. Ksir, “Decomposing representations of finite groups on Riemann-Roch spaces,” Proc. Amer. Math. Soc., to be published. [9] D. Joyner and W. Traves, “Representations of finite groups on Riemann-Roch spaces,” 2004 [Online]. Available: http://front.math.ucdavis.edu/math.AG/0210408, preprint [10] M. Kreuzer and L. Robbiano, Computational Commutative Algebra, I. New York: Springer-Verlag, 2000. [11] W. Stein and D. Joyner, “SAGE: System for algebra and geometry experimentation,” SIGSAM Comm. Computer Algebra vol. 39, no. 2, pp. 61–64, 2005 [Online]. Available: http://sage.scipy.org/ [Online]. Available: http://sage.sf.net/ [12] H. Stichtenoth, Algebraic Function Fields and Codes. New York: Springer-Verlag, 1993. [13] M.A. Tsfasman and S.G. Vlädut, Algebraic-Geometric Codes. Hingham, MA: Kluwer Academic, 1991. [14] S. Wesemeyer, “On the automorphism group of various Goppa codes,” IEEE Trans. Inf. Theory, vol. 44, pp. 630–643, Mar. 1998.
Improved Upper Bounds for Codes With Unequal Error Protection Shraga I. Bross, Member, IEEE, and Simon Litsyn, Senior Member, IEEE
Abstract—Asymptotic nonexistence bounds for unequal error protecting codes with two protection levels are considered. We show that the improved estimates on the possible distance distributions for codes may sometimes yield sharper upper bounds than the previously known ones on the higher significance protection level of both nonlinear and linear codes having two protection levels. Index Terms—Asymptotic bounds, linear codes, nonlinear codes, unequal error protection.
I. INTRODUCTION There are cases of data transmission in which the information is not of equal importance and thus should be protected unequally. Codes designed for unequal error protection (UEP) allocate the available channel redundancy between the different data importance levels such that in low signal-to-noise ratio (SNR) conditions at least the important information can be retrieved with acceptable quality. A natural approach to communicate while providing UEP is to use a time-sharing protocol. For example, supposing that two levels of UEP are required, the encoder divides the transmission time into two segments. In the first segment, it sends the less significant data using a less powerful code, while in the second segment it sends the most significant data using a more powerful code. However, Cover in his seminal contribution [5] demonstrated the surprising result that, in contrast to a time-sharing scheme, a “superimposed” code designed properly for UEP may improve the aggregate transmission rate. Henceforth, we will consider n-length (linear as well as nonlinear) block codes with two importance levels. In this case, each message
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block is divided into two parts: m = (m1 ; m2 ), m 2 M = f1; . . . ; 2k g, = 1; 2. That is, the data portion corresponding to m1 is represented by a binary k1 -tuple, while that corresponding to m2 is represented by a binary k2 -tuple, and the overall message space k1 + k2 . The fact is M = M1 2 M2 = f1; . . . ; 2k g, where k that the data represented by m1 is protected differently than that represented by m2 is made explicit by introducing the separation vector [7], [8]. Suppose that a binary n-length code C is used with some encoding function c(m) : M1 2 M2 ! C then the separation of the code s = (s1 ; s2 ) is defined as s
min
(m;m ):m = 6 m
( ( ) ( )); 0
dH c m ; c m
= 1; 2
where dH (a; b) denotes the Hamming distance between the binary n-sequences a and b. Furthermore, the transmission rates for the cork responding data significance levels are defined by R = n , = 1; 2. s s Such a code will be referred to as an R1 ; n ; R2 ; n UEP code. Constructions of linear codes having two or more levels of protection have been reported in [4], [9], [10], [14], [16], [18], and [19], whereas constructions of nonlinear codes having two levels of protection can be found in [8]. In [2], Bassalygo et al. present an asymptotic upper bound on the rates of two-level binary UEP codes, while Katsman presents in [11] an analogous result for linear codes. This work considers the asymptotic performance of codes having two levels of protection. The main contribution of this paper is a new asymptotic upper bound on the relative distance 2 = s2 =n when given the triple (R1 ; 1 = s1 =n; R2 ). The bound is based on the analysis of possible distance distributions of codes and in some cases it improves on the previous results for nonlinear as well as linear UEP codes. This paper is organized as follows. In Section II, we review the previous upper bounds for the classes of nonlinear and linear codes. Then, Section III presents our new bound for both classes of codes together with some numerical examples. II. PREVIOUS RESULTS Let E n be the space of binary n-length sequences with the Hamming metric, and for a sequence (alternatively a point) e 2 E n let Be (r) be the ball centered at e with radius r . For any set W E n , let 2r (W ) denote the r -neighborhood of W , namely the set of points that are at a distance not larger than r from W , i.e.,
2r (W ) fe 2 E n : d(e; W ) rg =
e2W
()
Be r :
Let A(n; d; r) be the maximum size of a binary n-length code with minimum distance d that is located within a ball of radius r , and let A(n; d; w ) be the maximum size of a binary n-length constant weight w code with minimum distance at least d. Since A(n; d; r ) (r + 1)maxi=1;...;r A(n; d; i) any asymptotic upper bound on A(n; d; w) can be used to estimate the asymptotic behavior of A(n; d; r). A. Nonlinear Codes
Manuscript received December 1, 2004; revised March 13, 2006. S. I. Bross is with the Department of Electrical Engineering, Technion–Israel Institute of Technology, Technion City, Haifa 32000, Israel (e-mail: shraga@ee. technion.ac.il). S. Litsyn is with the Department of Electrical Engineering-Systems, Tel-Aviv University, Ramat-Aviv 69978, Israel (e-mail: [email protected]). Communicated by G. Zémor, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2006.876353
Bassalygo et al. present in [2] an asymptotic upper bound on the rate-pair of any two-level binary UEP code C with separation s = (1 n; 2 n), where 2 > 1 . Their bounding technique is based on the following ingredients. 1 Any two-level UEP code C can be expressed as a union of 2k sets Xm , m2 2 M2 , each one being a code of size 2k and distance d1 = 1 n. Thus, denoting by T (d; N; r ) the minimum volume
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of the r -neighborhood of a size N codebook with distance d, it follows that
Consequently, using any upper bound on the size of a linear
(n 0 k ; k ; n) code, one obtains an upper bound on the size of any (n; k ; k ; n; n) linear code as follows. j2r (Xm )j 2R n T d ; 2R n ; r : (1) Proposition 2 [9], [11]: In any two-level binary linear UEP code C m 2M with separation s = ( n; n), where > , and lesser significance rate R p Since any point e 2 E n belongs to (at most) A(n; d ; 1 0 10u R (1 0 R ) 1 min 1 + H r) r -neighborhoods of the sets fXm g, it follows that 2 2 u 0 1
1
2
2
2
1
2
1
1
2
2
2M
j2r (Xm )j 2n A(n; d ; r): 2
2
1
2
2
m
2
1
1
0
1
2~
0 H 1=2 0 1 0
(2)
u2 + 2~2 u + 2~2 =2 (7)
Setting
2 =(1 0 R1 ). Here, the expression on the right-hand side with ~2 (r.h.s.) of (7) is a “projection” of the linear programming bound [17, eq. (1.4)].
lim inf n0 log T (n; 2Rn ; n) n!1 a(; ) lim sup n0 log A(n; n; n) n!1
(; R; )
1
1
2
2
the combination (1) and (2) yields the following asymptotic bound on R2 for a given triple (R1 ; 1 ; 2 ). Proposition 1 [2, Assert. 3]: In any two-level binary UEP code C with separation s = (1 n; 2 n), where 2 > 1 , and lesser significance rate R1
min f1 + a( ; ) 0 ( ; R ; )g p where %( ) = (1 0 1 0 2 )=2 is the Elias radius. R2
>%(
2
)
1
1
(3)
The bound (3) will be meaningful once a good upper bound on
a(2 ; ) and a good lower bound for (1 ; R1 ; ) are suggested. A tight upper bound on a(2 ; ) is the linear programming bound [17] which states that for any %( ) a(; ) ALP 1 (; ) H
1 (1 0 1 0 (0 + 2
2 0 2 + 4(1 0 ))2 )
III. NEW UPPER BOUND The analysis of possible distance distributions of codes provided significant improvements on the error exponents for maximum-likelihood decoding and error detecting over the binary symmetric channel, as evidenced in [15]. This approach hinges upon the nonnegativity of the MacWilliams transform of the distance distribution of a code [6], and on a proper choice of a suitable polynomial that satisfies some predefined boundary conditions. Let C be a code with a distance distribution B = (B0 ; B1 ; . . . ; Bn ), then as the block-length becomes large enough the components of the distance distribution are lower bounded as follows. Proposition 3 [15, Th. 6]: Let C be a code with rate R(C) and sufficiently large block-length n. For any H 01 (R(C)), there exists a component Bn of the distance distribution such that
n01 log2 Bn R(C) + H ( ) 0 2p( ; )
(8)
(4) where H (x) = 0x log2 x 0 (10 x)log2 (10 x) is the binary entropy function. However, in order to obtain a meaningful lower bound for (1 ; R1 ; ) the authors of [2] suggest splitting the evaluation of the n-neighborhood into two steps. First, they estimate the n-neighborhood of the original code (0 ). Then, the remaining ( 0 )n-neighborhood to yield (1 ; R1 ; ) (0; (1 ; R1 ; ); 0 ) 0 : (5) Having done so, substitution of the obvious bound (1 ; R1 ; ) R1 + H ( ) 0 a(1 ; ) in (5) yields the lower bound (see [12])
( 1 ; R 1 ; )
max H (minf 0 + ; 1=2g)
%(
)
(6)
where 1=2 is the solution of H ( ) = R1 + H ( ) 0 a(1 ; ). Finally, for small values of R1 , that is, when R1 H (2 ) 0 1 0 (1 0 1 )H 100 =2 , the expression (1 ; R1 ; 2 ) = R1 + H (2 ) should be used in (3). B. Linear Codes Katsman presents in [11] an asymptotic upper bound on the ratepair of any two-level binary linear UEP code C with separation s = (1 n; 2 n), where 2 > 1 . His bound is based on the observation that the existence of a linear (n; k1 ; k2 ; 1 n; 2 n) two-level UEP code implies the existence of a linear (n 0 k1 ; k2 ) code with minimum distance 2 n .
where
p( ; )
H ( )
+
0
0 2 ) 0 4y(1 0 y) log (1 0 2 ) + (1 2(1 0 y) 2
2
dy
and 2 (0; 1=20 (1 0 )]. Here, f g means f g (1+ on (1)) and we write on (1) to denote an unspecified real-valued function that goes to zero as n tends to infinity. A. Nonlinear Codes Consider a binary two-level (R1 ; 1 ; R2 ; 2 ) UEP code C . On the one hand, C may be viewed as a collection of 2(R +R )n codewords having a minimal spectral component of relative distance 1 at a rate not larger than R1 . However, C may be viewed as well as a collection of 2R n subcodes each of rate R1 and minimum relative distance 1 , while the minimum relative distance between any two codewords belonging to different subcodes is at least 2 . As long as one provides a firm lower bound on the distance distribution of the code C and this lower bound lies above any upper bound on the distance distribution of an (R1 ; 1 ) subcode, it must be that at least one of the spectral components which arise while considering the complete collection of 2R n subcodes must lie above that lower bound. However, when considering the complete collection of 2R n subcodes the first spectral component that can be larger than that of the (R1 ; 1 ) subcode is precisely the higher significance level 2 . Proposition 3 can now be used in order to upper bound 2 for a given triple (R1 ; 1 ; R2 ) as follows.
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Proposition 4: Let A (R1 ; 1 ); 1 be the best upper bound on the distance distribution of some subcode Xm of C of rate R1 and relative minimum distance 1 , and let 3 2 (0; H 01 (R1 + R2 )) be the argument that minimizes the vertical distance between the lower bound (8), evaluated with R(C ) = R1 + R2 , and A (R1 ; 1 ); that is,
3 = arg
( ; ) min 2(0;H p (R +R )] 2(0;1=20 (10 )] ( ;)>0
3)
A (R 1 ; 1 )
min 2[0;1=2] 2[0;1=2] 2(0;1]
0 1 0p p (1 0 R ) + 1 01 p H ( ) 1 0 2p minfALS (; ); ALP (; )g + 10p 0 h(; ; ) 0 log p 0 (1 0 ) log (1 0 p) h(; ; ) = max H + (1 0 )H 0 10 0 + + H + (1 0 )H 10 1
1
2
where
( ; ) R1 + R2 + H ( ) 0 2p( ; ) 0 A (R1 ; 1 ):
2
1
1
2
Then
2 0:5 0 3 (1 0 3 ):
(9)
Proof: The interpretation of Proposition 3 in our case is as follows. The UEP code has rate R(C ) = R1 + R2 , therefore for an arbitrary choice of 2 0; H 01 (R1 + R2 ) the r.h.s. of (8) provides a lower bound on the distance distribution of the UEP code in the range 2 (0; 1=2 0 (1 0 )] in the sense that, at least one component of the distance distribution of the UEP code must lie above the curve (8) in that range. Let us denote such a lower bound, which traces the r.h.s. of (8) in the range 2 (0; 1=2 0 (1 0 )], as a '( ) curve. On the other hand, the UEP code has 2R n subcodes each of rate R1 and minimum relative distance 1 . Consequently, the spectral component B(C) of the code C is
BC
( )
= 20(R +R )n = 20R n
c2C c 2C
c;c 2X
1fd(c;c )= ng
(X 1fd(c;c )= ng = B
)
(10)
since none of the codeword pairs belonging to different subcodes contributes to the first summation in (10). Let A (R1 ; 1 ) denote the best known upper bound on the distance distribution of any such subcode. If (R1 ; 1 ; R2 ; 2 ) is an achievable quadruple for an UEP code and the '( ) curve “passes” above A (R1 ; 1 ) such that the vertical distance between the two is always positive it must be that 2 is smaller than or equal to the maximum (i.e., rightmost) for which the curve '( ) has been evaluated. Otherwise, we arrive at a contradiction since, by our choice of , none of the distance distribution components of any rate R1 subcode appears to satisfy the lower bound (8), whereas the lowest component in the distance distribution of the UEP code which might be larger than that of the (R1 ; 1 ) subcode is 2 . The claim is proved. The bound (9) will be meaningful once a strong upper bound on A (R1 ; 1 ) is provided. The following Lemma summarizes the best known upper bound on A (R1 ; 1 ). Lemma 1: The distance distribution of a code C of rate R1 and minimum relative distance 1 is bounded from above as follows. 1)
A (R1 ; 1 ) ALP 1 (; ); 2)
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0
(1 0 ):
A (R1 ; 1 ) ALS (; ) H () 0 H (0 ) 0 1=2 + ALP 1 (; 0 ); 0 = arg min 1 0 H ( ) + ALP 1 (; ): %() 1=2
2
where the maximum is taken over in the interval (maxf0; 0
; 0 g; minf1; + ; + g). Finally, (1 ; 2 ) are defined as
1 = 2 =
; if + 0 2 ( + 0 )=2; otherwise ( 0 + )=2; if 2 (1 0 ) > 0 + (1 0 ); otherwise.
The first bound is the linear programming bound for constant weight codes [17] which is the best known for large distances. The second bound is due to a recurrence relation developed by Levenshtein [13] and is given in the form suggested by Samorodnitsky [20]. The third bound which takes into account the known size of the code is derived in [1, Th. 7] based on Beckner’s inequality [3]. Perhaps the easiest way to demonstrate Proposition 4 is by taking A (R1 ; 1 ) to be the step function (i.e., the weakest possible bound in Lemma 1)
0;
< 1 R 1 ; 1 .
A (R 1 ; 1 ) = With this choice, let 3 satisfies
2 (0; H 0 (R 1
1
+ R2 )] be the argument that
R1 + R2 + H ( 3 ) 0 2p( 3 ; 1 ) = R1 + ;
> 0:
Then, (9) holds and the tightest bound is obtained as ! 0. Example 1: Let 1 = 0:07, 2 = 0:2 then, for any given R1 , inequality (3) provides an upper bound on the attainable rate R2 . For any possible triple (R1 ; 1 = 0:07; R2 ) as per Proposition 1, Fig. 1 illustrates the upper bound on 2 according to Proposition 3. It can be seen that in the range 0:014 R1 0:08 the possible relative distance 2 is strictly smaller than 0.2 thereby sharpening the upper bound provided by (3). In the range 0:053 R1 0:08 the bound is obtained with a lower bound on the distance distribution [as per (8)] which “touches” the upper bound A (R1 ; 1 ) whereas in the range R1 < 0:053 the curve corresponding to 3 = H 01 (R1 + R2 ) already passes fairly above A (R1 ; 1 ). These observations explain the behavior of the bound in Fig. 1. B. Linear Codes Following the same arguments as in Section III-A, Proposition 3 may be used in order to upper bound 2 when given a triple (R1 ; 1 ; R2 ) as follows.
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Fig. 1. Upper bound on for optimal (R ; R ) pairs for a given . TABLE I UPPER BOUND ON , AS PER PROPOSITION 5, WHEN GIVEN (R ; ; R )
(L )
Proposition 5: Let A (R1 ; 1 ), 1 be the best upper bound on the distance distribution of some linear subcode Xm of C of rate R1 and relative minimum distance 1 , and let 3 2 0; H 01 (R1 + R2 ) be the argument that minimizes the vertical distance between the lower (L) bound (8), evaluated with R(C ) = R1 + R2 , and A (R1 ; 1 ); that is
3 = arg
( ; ) min 2(0;H p (R +R )] 2(0;1=20 (10 )] ( ;)>0
where
b(L) (R1 ; 1 ) min0 1
R1 H ( )
+ (1 0 )R1
(1 0 )R1
1
R1 ; Here, 3 ( ) is the root of the equation
( ; ) R1 + R2 + H ( ) 0 2p( ; ) 0 A(L) (R1 ; 1 ): Then, 2 is upper bounded by (9). Again, this bound will be meaningful once a good upper bound on A(L) (R1 ; 1 ) is provided. The following lemma summarizes the best (L) known upper bound on A (R1 ; 1 ). Lemma 2 [1, Th. 8]: The distance distribution of a linear code C of rate R1 and minimum relative distance 1 is bounded from above as follows:
A(L) (R1 ; 1 ) min A (R1 ; 1 ); b(L) (R1 ; 1 )
where
(11)
= R3
; 3 ( ) > 2 3 ( ) 2 .
and R3 ( ) is any upper bound on the rate of a code with minimum distance n (e.g., the linear programming bound). Example 2: Let 1 = 0:01, 2 = 0:3. Then, for any given R1 , inequality (7) provides an upper bound on the attainable rate R2 . According to Proposition 2, the rate pair (R1 = 0:003; R2 = 0:2477) is attainable. However, based on Proposition 5, one may verify that for the triple (R1 = 0:003; 1 = 0:01; R2 = 0:3477) it must be that 2 < 0:2999. Table I presents a few more combinations of (R1 ; 1 ; R2 ) for which the bound of Proposition 5 improves upon that of (7). This example suggests that the upper bound for linear UEP codes that hinges upon [11, Lemma 2] is far from being tight since a bound on the distance distribution of nonlinear codes already improves upon it.
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Capacity Achieving Code Constructions for Two Classes Constraints of
ACKNOWLEDGMENT The authors would like to thank the reviewers for their careful reading of the manuscript and their constructive comments.
REFERENCES [1] A. Ashikhmin, G. Cohen, M. Krivelevich, and S. Litsyn, “Bounds on distance distribution in codes of known size,” IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 259–264, Jan. 2005. [2] L. A. Bassalygo, V. A. Zinov’ev, V. V. Zyablov, M. S. Pinsker, and G. S. Poltyrev, “Bounds for codes with unequal protection of two sets of messages,” Probl. Peredach. Inf., vol. 15, no. 3, pp. 40–49, Jul.–Sep. 1979. [3] W. Beckner, “Inequalities in Fourier analysis,” Ann. Math., vol. 102, pp. 159–172, 1975. [4] I. M. Boyarinov and G. L. Katsman, “Linear unequal error protection codes,” IEEE Trans. Inf. Theory, vol. IT-27, no. 3, pp. 168–175, Mar. 1981. [5] T. M. Cover, “Broadcast channels,” IEEE Trans. Inf. Theory, vol. IT-18, no. 1, pp. 2–14, Jan. 1972. [6] P. Delsarte, “An algebraic approach to the association schemes of coding theory,” Philips Res. Rep. Suppl., no. 10, pp. 1–97, 1973. [7] L. A. Dunning and W. E. Robbins, “Optimal encoding of linear block codes for enequal error protection,” Inf. Control, vol. 37, pp. 150–177, 1978. [8] E. K. Englund, “Nonlinear unequal error protection codes are sometimes better than linear ones,” IEEE Trans. Inf. Theory, vol. 37, no. 9, pp. 1418–1420, Sep. 1991. [9] E. K. Englund and A. I. Hansson, “Constructive codes with unequal error protection,” IEEE Trans. Inf. Theory, vol. 43, no. 3, pp. 715–721, Mar. 1997. [10] W. J. van Gils, “Two topics on linear unequal error protection codes: Bounds and their lengths and cyclic code classes,” IEEE Trans. Inf. Theory, vol. IT-29, no. 9, pp. 866–876, Sep. 1983. [11] G. L. Katsman, “Bounds on volume of linear codes with unequal information-symbol protection,” Probl. Peredach. Inf., vol. 16, no. 2, pp. 25–32, Apr.–Jun. 1980. [12] G. O. H. Katona, “The Hamming sphere has minimum boundary,” Stud. Sci. Math. Hungarica, vol. 10, pp. 131–140, 1975. [13] V. I. Levenshtein, “Upper-bound estimates for fixed-weight codes,” Probl. Peredach. Inf., vol. 7, no. 4, pp. 3–12, Oct.–Dec. 1971. [14] M. C. Lin and S. Lin, “Codes with multi-level error correcting capabilities,” Discrete Math., vol. 83, pp. 301–314, 1990. [15] S. Litsyn, “New upper bounds on error exponents,” IEEE Trans. Inf. Theory, vol. 45, no. 3, pp. 385–398, Mar. 1999. [16] B. Masnick and J. K. Wolf, “On linear unequal error protection codes,” IEEE Trans. Inf. Theory, vol. IT-3, no. 10, pp. 600–607, Oct. 1967. [17] R. J. McEliece, E. R. Rodemich, H. Rumsey, Jr, and L. R. Welch, “New upper bounds on the rate of a code via the Delsarte–MacWilliams inequalities,” IEEE Trans. Inf. Theory, vol. IT-23, no. 3, pp. 157–166, Mar. 1977. [18] R. H. Morelos-Zaragoza and S. Lin, “On a class of optimal nonbinary linear unequal error protection codes for two sets of messages,” IEEE Trans. Inf. Theory, vol. 40, no. 1, pp. 196–200, Jan. 1994. [19] F.Özbudak and H. Stichtenoth, “Constructing linear unequal error protection codes from algebraic curves,” IEEE Trans. Inf. Theory, vol. 49, no. 6, pp. 1523–1527, Jun. 2003. [20] A. Samorodnitsky, “On the optimum of Delsarte’s linear program,” J. Combin. Theory, ser. A, vol. 96, no. 2, pp. 261–287, 2001.
Yogesh Sankarasubramaniam, Student Member, IEEE, and Steven W. McLaughlin, Fellow, IEEE
Abstract—This correspondence presents two variable-rate encoding al) constraint when = 2 +1, gorithms that achieve capacity for the ( +1 is not prime. The first algorithm, symbol sliding, is a genor when eralized version of the bit flipping algorithm introduced by Aviran et al. In addition to achieving capacity for ( 2 +1) constraints, it comes close to capacity in other cases. The second algorithm is based on interleaving and is a generalized version of the bit stuffing algorithm introduced by Bender and Wolf. This method uses fewer than biased bit streams to achieve ) constraints with + 1 not prime. In particular, capacity for ( +2 1) constraints 2 requires the encoder for ( only biased bit streams. )-constrained sequences,
Index Terms—Bit flipping, bit stuffing, ( Shannon capacity.
I. INTRODUCTION A binary sequence is said to be (d; k)-constrained if successive “1”s are separated by at least d and at most k consecutive “0”s. A (d; k) code is defined as an invertible mapping between unconstrained binary sequences and (d; k)-constrained sequences. There is a long history of (d; k ) codes, and they are part of virtually all magnetic and optical disc recording systems today (see [4] for an overview). The d constraint is used to regulate intersymbol interference and the k constraint is important for timing recovery. Let us denote by R(d; k), the encoding rate of a (d; k) code. It is well known that the maximum possible value of R(d; k) is equal to the Shannon capacity of the (d; k) constraint given by [8]
C (d; k) = log2 d;k ; where d;k is the positive, real root of the characteristic equation Gd;k (z ) = 1, and Gd;k (z ) is the characteristic polynomial of the (d; k ) constraint, given by
Gd;k (z ) =
k+1
j =d+1
z 0j
z 01 + z 0(d+1)
when k < 1 when k = 1:
(1)
R(d;k) measures how close the The encoding efficiency E (d; k) = C (d;k) code is to capacity. A (d; k) code is said to be optimal if it is 100% efficient. In this correspondence, we present optimal code constructions for two classes of (d; k) constraints. The proposed codes are variable-rate, in the sense that for any given finite input block length, the generated output block length can vary depending on the actual input bits. In [6] and [7], optimal (d; k) codes have been studied from a source coding perspective. This can be understood as follows. For given nonnegative integers d and k; 0 d < k , let Xd;k = f0k 1; 0k01 1; . . . ; 0d+1 1; 0d 1g if k < 1, and let
Manuscript received June 21, 2004; revised February 17, 2006. This work was supported by Seagate Research. The material in this correspondence was presented in part at the International Symposium on Information Theory and Its Applications, Parma, Italy, October 2004. The authors are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: yogi@ece. gatech.edu; [email protected]). Communicated by K. A. S. Abdel-Ghaffar, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2006.876224 0018-9448/$20.00 © 2006 IEEE
[8] D. Joyner and A. Ksir, “Decomposing representations of finite groups on Riemann-Roch spaces,” Proc. Amer. Math. Soc., to be published. [9] D. Joyner and W. Traves, “Representations of finite groups on Riemann-Roch spaces,” 2004 [Online]. Available: http://front.math.ucdavis.edu/math.AG/0210408, preprint [10] M. Kreuzer and L. Robbiano, Computational Commutative Algebra, I. New York: Springer-Verlag, 2000. [11] W. Stein and D. Joyner, “SAGE: System for algebra and geometry experimentation,” SIGSAM Comm. Computer Algebra vol. 39, no. 2, pp. 61–64, 2005 [Online]. Available: http://sage.scipy.org/ [Online]. Available: http://sage.sf.net/ [12] H. Stichtenoth, Algebraic Function Fields and Codes. New York: Springer-Verlag, 1993. [13] M.A. Tsfasman and S.G. Vlädut, Algebraic-Geometric Codes. Hingham, MA: Kluwer Academic, 1991. [14] S. Wesemeyer, “On the automorphism group of various Goppa codes,” IEEE Trans. Inf. Theory, vol. 44, pp. 630–643, Mar. 1998.
Improved Upper Bounds for Codes With Unequal Error Protection Shraga I. Bross, Member, IEEE, and Simon Litsyn, Senior Member, IEEE
Abstract—Asymptotic nonexistence bounds for unequal error protecting codes with two protection levels are considered. We show that the improved estimates on the possible distance distributions for codes may sometimes yield sharper upper bounds than the previously known ones on the higher significance protection level of both nonlinear and linear codes having two protection levels. Index Terms—Asymptotic bounds, linear codes, nonlinear codes, unequal error protection.
I. INTRODUCTION There are cases of data transmission in which the information is not of equal importance and thus should be protected unequally. Codes designed for unequal error protection (UEP) allocate the available channel redundancy between the different data importance levels such that in low signal-to-noise ratio (SNR) conditions at least the important information can be retrieved with acceptable quality. A natural approach to communicate while providing UEP is to use a time-sharing protocol. For example, supposing that two levels of UEP are required, the encoder divides the transmission time into two segments. In the first segment, it sends the less significant data using a less powerful code, while in the second segment it sends the most significant data using a more powerful code. However, Cover in his seminal contribution [5] demonstrated the surprising result that, in contrast to a time-sharing scheme, a “superimposed” code designed properly for UEP may improve the aggregate transmission rate. Henceforth, we will consider n-length (linear as well as nonlinear) block codes with two importance levels. In this case, each message
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block is divided into two parts: m = (m1 ; m2 ), m 2 M = f1; . . . ; 2k g, = 1; 2. That is, the data portion corresponding to m1 is represented by a binary k1 -tuple, while that corresponding to m2 is represented by a binary k2 -tuple, and the overall message space k1 + k2 . The fact is M = M1 2 M2 = f1; . . . ; 2k g, where k that the data represented by m1 is protected differently than that represented by m2 is made explicit by introducing the separation vector [7], [8]. Suppose that a binary n-length code C is used with some encoding function c(m) : M1 2 M2 ! C then the separation of the code s = (s1 ; s2 ) is defined as s
min
(m;m ):m = 6 m
( ( ) ( )); 0
dH c m ; c m
= 1; 2
where dH (a; b) denotes the Hamming distance between the binary n-sequences a and b. Furthermore, the transmission rates for the cork responding data significance levels are defined by R = n , = 1; 2. s s Such a code will be referred to as an R1 ; n ; R2 ; n UEP code. Constructions of linear codes having two or more levels of protection have been reported in [4], [9], [10], [14], [16], [18], and [19], whereas constructions of nonlinear codes having two levels of protection can be found in [8]. In [2], Bassalygo et al. present an asymptotic upper bound on the rates of two-level binary UEP codes, while Katsman presents in [11] an analogous result for linear codes. This work considers the asymptotic performance of codes having two levels of protection. The main contribution of this paper is a new asymptotic upper bound on the relative distance 2 = s2 =n when given the triple (R1 ; 1 = s1 =n; R2 ). The bound is based on the analysis of possible distance distributions of codes and in some cases it improves on the previous results for nonlinear as well as linear UEP codes. This paper is organized as follows. In Section II, we review the previous upper bounds for the classes of nonlinear and linear codes. Then, Section III presents our new bound for both classes of codes together with some numerical examples. II. PREVIOUS RESULTS Let E n be the space of binary n-length sequences with the Hamming metric, and for a sequence (alternatively a point) e 2 E n let Be (r) be the ball centered at e with radius r . For any set W E n , let 2r (W ) denote the r -neighborhood of W , namely the set of points that are at a distance not larger than r from W , i.e.,
2r (W ) fe 2 E n : d(e; W ) rg =
e2W
()
Be r :
Let A(n; d; r) be the maximum size of a binary n-length code with minimum distance d that is located within a ball of radius r , and let A(n; d; w ) be the maximum size of a binary n-length constant weight w code with minimum distance at least d. Since A(n; d; r ) (r + 1)maxi=1;...;r A(n; d; i) any asymptotic upper bound on A(n; d; w) can be used to estimate the asymptotic behavior of A(n; d; r). A. Nonlinear Codes
Manuscript received December 1, 2004; revised March 13, 2006. S. I. Bross is with the Department of Electrical Engineering, Technion–Israel Institute of Technology, Technion City, Haifa 32000, Israel (e-mail: shraga@ee. technion.ac.il). S. Litsyn is with the Department of Electrical Engineering-Systems, Tel-Aviv University, Ramat-Aviv 69978, Israel (e-mail: [email protected]). Communicated by G. Zémor, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2006.876353
Bassalygo et al. present in [2] an asymptotic upper bound on the rate-pair of any two-level binary UEP code C with separation s = (1 n; 2 n), where 2 > 1 . Their bounding technique is based on the following ingredients. 1 Any two-level UEP code C can be expressed as a union of 2k sets Xm , m2 2 M2 , each one being a code of size 2k and distance d1 = 1 n. Thus, denoting by T (d; N; r ) the minimum volume
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of the r -neighborhood of a size N codebook with distance d, it follows that
Consequently, using any upper bound on the size of a linear
(n 0 k ; k ; n) code, one obtains an upper bound on the size of any (n; k ; k ; n; n) linear code as follows. j2r (Xm )j 2R n T d ; 2R n ; r : (1) Proposition 2 [9], [11]: In any two-level binary linear UEP code C m 2M with separation s = ( n; n), where > , and lesser significance rate R p Since any point e 2 E n belongs to (at most) A(n; d ; 1 0 10u R (1 0 R ) 1 min 1 + H r) r -neighborhoods of the sets fXm g, it follows that 2 2 u 0 1
1
2
2
2
1
2
1
1
2
2
2M
j2r (Xm )j 2n A(n; d ; r): 2
2
1
2
2
m
2
1
1
0
1
2~
0 H 1=2 0 1 0
(2)
u2 + 2~2 u + 2~2 =2 (7)
Setting
2 =(1 0 R1 ). Here, the expression on the right-hand side with ~2 (r.h.s.) of (7) is a “projection” of the linear programming bound [17, eq. (1.4)].
lim inf n0 log T (n; 2Rn ; n) n!1 a(; ) lim sup n0 log A(n; n; n) n!1
(; R; )
1
1
2
2
the combination (1) and (2) yields the following asymptotic bound on R2 for a given triple (R1 ; 1 ; 2 ). Proposition 1 [2, Assert. 3]: In any two-level binary UEP code C with separation s = (1 n; 2 n), where 2 > 1 , and lesser significance rate R1
min f1 + a( ; ) 0 ( ; R ; )g p where %( ) = (1 0 1 0 2 )=2 is the Elias radius. R2
>%(
2
)
1
1
(3)
The bound (3) will be meaningful once a good upper bound on
a(2 ; ) and a good lower bound for (1 ; R1 ; ) are suggested. A tight upper bound on a(2 ; ) is the linear programming bound [17] which states that for any %( ) a(; ) ALP 1 (; ) H
1 (1 0 1 0 (0 + 2
2 0 2 + 4(1 0 ))2 )
III. NEW UPPER BOUND The analysis of possible distance distributions of codes provided significant improvements on the error exponents for maximum-likelihood decoding and error detecting over the binary symmetric channel, as evidenced in [15]. This approach hinges upon the nonnegativity of the MacWilliams transform of the distance distribution of a code [6], and on a proper choice of a suitable polynomial that satisfies some predefined boundary conditions. Let C be a code with a distance distribution B = (B0 ; B1 ; . . . ; Bn ), then as the block-length becomes large enough the components of the distance distribution are lower bounded as follows. Proposition 3 [15, Th. 6]: Let C be a code with rate R(C) and sufficiently large block-length n. For any H 01 (R(C)), there exists a component Bn of the distance distribution such that
n01 log2 Bn R(C) + H ( ) 0 2p( ; )
(8)
(4) where H (x) = 0x log2 x 0 (10 x)log2 (10 x) is the binary entropy function. However, in order to obtain a meaningful lower bound for (1 ; R1 ; ) the authors of [2] suggest splitting the evaluation of the n-neighborhood into two steps. First, they estimate the n-neighborhood of the original code (0 ). Then, the remaining ( 0 )n-neighborhood to yield (1 ; R1 ; ) (0; (1 ; R1 ; ); 0 ) 0 : (5) Having done so, substitution of the obvious bound (1 ; R1 ; ) R1 + H ( ) 0 a(1 ; ) in (5) yields the lower bound (see [12])
( 1 ; R 1 ; )
max H (minf 0 + ; 1=2g)
%(
)
(6)
where 1=2 is the solution of H ( ) = R1 + H ( ) 0 a(1 ; ). Finally, for small values of R1 , that is, when R1 H (2 ) 0 1 0 (1 0 1 )H 100 =2 , the expression (1 ; R1 ; 2 ) = R1 + H (2 ) should be used in (3). B. Linear Codes Katsman presents in [11] an asymptotic upper bound on the ratepair of any two-level binary linear UEP code C with separation s = (1 n; 2 n), where 2 > 1 . His bound is based on the observation that the existence of a linear (n; k1 ; k2 ; 1 n; 2 n) two-level UEP code implies the existence of a linear (n 0 k1 ; k2 ) code with minimum distance 2 n .
where
p( ; )
H ( )
+
0
0 2 ) 0 4y(1 0 y) log (1 0 2 ) + (1 2(1 0 y) 2
2
dy
and 2 (0; 1=20 (1 0 )]. Here, f g means f g (1+ on (1)) and we write on (1) to denote an unspecified real-valued function that goes to zero as n tends to infinity. A. Nonlinear Codes Consider a binary two-level (R1 ; 1 ; R2 ; 2 ) UEP code C . On the one hand, C may be viewed as a collection of 2(R +R )n codewords having a minimal spectral component of relative distance 1 at a rate not larger than R1 . However, C may be viewed as well as a collection of 2R n subcodes each of rate R1 and minimum relative distance 1 , while the minimum relative distance between any two codewords belonging to different subcodes is at least 2 . As long as one provides a firm lower bound on the distance distribution of the code C and this lower bound lies above any upper bound on the distance distribution of an (R1 ; 1 ) subcode, it must be that at least one of the spectral components which arise while considering the complete collection of 2R n subcodes must lie above that lower bound. However, when considering the complete collection of 2R n subcodes the first spectral component that can be larger than that of the (R1 ; 1 ) subcode is precisely the higher significance level 2 . Proposition 3 can now be used in order to upper bound 2 for a given triple (R1 ; 1 ; R2 ) as follows.
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Proposition 4: Let A (R1 ; 1 ); 1 be the best upper bound on the distance distribution of some subcode Xm of C of rate R1 and relative minimum distance 1 , and let 3 2 (0; H 01 (R1 + R2 )) be the argument that minimizes the vertical distance between the lower bound (8), evaluated with R(C ) = R1 + R2 , and A (R1 ; 1 ); that is,
3 = arg
( ; ) min 2(0;H p (R +R )] 2(0;1=20 (10 )] ( ;)>0
3)
A (R 1 ; 1 )
min 2[0;1=2] 2[0;1=2] 2(0;1]
0 1 0p p (1 0 R ) + 1 01 p H ( ) 1 0 2p minfALS (; ); ALP (; )g + 10p 0 h(; ; ) 0 log p 0 (1 0 ) log (1 0 p) h(; ; ) = max H + (1 0 )H 0 10 0 + + H + (1 0 )H 10 1
1
2
where
( ; ) R1 + R2 + H ( ) 0 2p( ; ) 0 A (R1 ; 1 ):
2
1
1
2
Then
2 0:5 0 3 (1 0 3 ):
(9)
Proof: The interpretation of Proposition 3 in our case is as follows. The UEP code has rate R(C ) = R1 + R2 , therefore for an arbitrary choice of 2 0; H 01 (R1 + R2 ) the r.h.s. of (8) provides a lower bound on the distance distribution of the UEP code in the range 2 (0; 1=2 0 (1 0 )] in the sense that, at least one component of the distance distribution of the UEP code must lie above the curve (8) in that range. Let us denote such a lower bound, which traces the r.h.s. of (8) in the range 2 (0; 1=2 0 (1 0 )], as a '( ) curve. On the other hand, the UEP code has 2R n subcodes each of rate R1 and minimum relative distance 1 . Consequently, the spectral component B(C) of the code C is
BC
( )
= 20(R +R )n = 20R n
c2C c 2C
c;c 2X
1fd(c;c )= ng
(X 1fd(c;c )= ng = B
)
(10)
since none of the codeword pairs belonging to different subcodes contributes to the first summation in (10). Let A (R1 ; 1 ) denote the best known upper bound on the distance distribution of any such subcode. If (R1 ; 1 ; R2 ; 2 ) is an achievable quadruple for an UEP code and the '( ) curve “passes” above A (R1 ; 1 ) such that the vertical distance between the two is always positive it must be that 2 is smaller than or equal to the maximum (i.e., rightmost) for which the curve '( ) has been evaluated. Otherwise, we arrive at a contradiction since, by our choice of , none of the distance distribution components of any rate R1 subcode appears to satisfy the lower bound (8), whereas the lowest component in the distance distribution of the UEP code which might be larger than that of the (R1 ; 1 ) subcode is 2 . The claim is proved. The bound (9) will be meaningful once a strong upper bound on A (R1 ; 1 ) is provided. The following Lemma summarizes the best known upper bound on A (R1 ; 1 ). Lemma 1: The distance distribution of a code C of rate R1 and minimum relative distance 1 is bounded from above as follows. 1)
A (R1 ; 1 ) ALP 1 (; ); 2)
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0
(1 0 ):
A (R1 ; 1 ) ALS (; ) H () 0 H (0 ) 0 1=2 + ALP 1 (; 0 ); 0 = arg min 1 0 H ( ) + ALP 1 (; ): %() 1=2
2
where the maximum is taken over in the interval (maxf0; 0
; 0 g; minf1; + ; + g). Finally, (1 ; 2 ) are defined as
1 = 2 =
; if + 0 2 ( + 0 )=2; otherwise ( 0 + )=2; if 2 (1 0 ) > 0 + (1 0 ); otherwise.
The first bound is the linear programming bound for constant weight codes [17] which is the best known for large distances. The second bound is due to a recurrence relation developed by Levenshtein [13] and is given in the form suggested by Samorodnitsky [20]. The third bound which takes into account the known size of the code is derived in [1, Th. 7] based on Beckner’s inequality [3]. Perhaps the easiest way to demonstrate Proposition 4 is by taking A (R1 ; 1 ) to be the step function (i.e., the weakest possible bound in Lemma 1)
0;
< 1 R 1 ; 1 .
A (R 1 ; 1 ) = With this choice, let 3 satisfies
2 (0; H 0 (R 1
1
+ R2 )] be the argument that
R1 + R2 + H ( 3 ) 0 2p( 3 ; 1 ) = R1 + ;
> 0:
Then, (9) holds and the tightest bound is obtained as ! 0. Example 1: Let 1 = 0:07, 2 = 0:2 then, for any given R1 , inequality (3) provides an upper bound on the attainable rate R2 . For any possible triple (R1 ; 1 = 0:07; R2 ) as per Proposition 1, Fig. 1 illustrates the upper bound on 2 according to Proposition 3. It can be seen that in the range 0:014 R1 0:08 the possible relative distance 2 is strictly smaller than 0.2 thereby sharpening the upper bound provided by (3). In the range 0:053 R1 0:08 the bound is obtained with a lower bound on the distance distribution [as per (8)] which “touches” the upper bound A (R1 ; 1 ) whereas in the range R1 < 0:053 the curve corresponding to 3 = H 01 (R1 + R2 ) already passes fairly above A (R1 ; 1 ). These observations explain the behavior of the bound in Fig. 1. B. Linear Codes Following the same arguments as in Section III-A, Proposition 3 may be used in order to upper bound 2 when given a triple (R1 ; 1 ; R2 ) as follows.
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Fig. 1. Upper bound on for optimal (R ; R ) pairs for a given . TABLE I UPPER BOUND ON , AS PER PROPOSITION 5, WHEN GIVEN (R ; ; R )
(L )
Proposition 5: Let A (R1 ; 1 ), 1 be the best upper bound on the distance distribution of some linear subcode Xm of C of rate R1 and relative minimum distance 1 , and let 3 2 0; H 01 (R1 + R2 ) be the argument that minimizes the vertical distance between the lower (L) bound (8), evaluated with R(C ) = R1 + R2 , and A (R1 ; 1 ); that is
3 = arg
( ; ) min 2(0;H p (R +R )] 2(0;1=20 (10 )] ( ;)>0
where
b(L) (R1 ; 1 ) min0 1
R1 H ( )
+ (1 0 )R1
(1 0 )R1
1
R1 ; Here, 3 ( ) is the root of the equation
( ; ) R1 + R2 + H ( ) 0 2p( ; ) 0 A(L) (R1 ; 1 ): Then, 2 is upper bounded by (9). Again, this bound will be meaningful once a good upper bound on A(L) (R1 ; 1 ) is provided. The following lemma summarizes the best (L) known upper bound on A (R1 ; 1 ). Lemma 2 [1, Th. 8]: The distance distribution of a linear code C of rate R1 and minimum relative distance 1 is bounded from above as follows:
A(L) (R1 ; 1 ) min A (R1 ; 1 ); b(L) (R1 ; 1 )
where
(11)
= R3
; 3 ( ) > 2 3 ( ) 2 .
and R3 ( ) is any upper bound on the rate of a code with minimum distance n (e.g., the linear programming bound). Example 2: Let 1 = 0:01, 2 = 0:3. Then, for any given R1 , inequality (7) provides an upper bound on the attainable rate R2 . According to Proposition 2, the rate pair (R1 = 0:003; R2 = 0:2477) is attainable. However, based on Proposition 5, one may verify that for the triple (R1 = 0:003; 1 = 0:01; R2 = 0:3477) it must be that 2 < 0:2999. Table I presents a few more combinations of (R1 ; 1 ; R2 ) for which the bound of Proposition 5 improves upon that of (7). This example suggests that the upper bound for linear UEP codes that hinges upon [11, Lemma 2] is far from being tight since a bound on the distance distribution of nonlinear codes already improves upon it.
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Capacity Achieving Code Constructions for Two Classes Constraints of
ACKNOWLEDGMENT The authors would like to thank the reviewers for their careful reading of the manuscript and their constructive comments.
REFERENCES [1] A. Ashikhmin, G. Cohen, M. Krivelevich, and S. Litsyn, “Bounds on distance distribution in codes of known size,” IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 259–264, Jan. 2005. [2] L. A. Bassalygo, V. A. Zinov’ev, V. V. Zyablov, M. S. Pinsker, and G. S. Poltyrev, “Bounds for codes with unequal protection of two sets of messages,” Probl. Peredach. Inf., vol. 15, no. 3, pp. 40–49, Jul.–Sep. 1979. [3] W. Beckner, “Inequalities in Fourier analysis,” Ann. Math., vol. 102, pp. 159–172, 1975. [4] I. M. Boyarinov and G. L. Katsman, “Linear unequal error protection codes,” IEEE Trans. Inf. Theory, vol. IT-27, no. 3, pp. 168–175, Mar. 1981. [5] T. M. Cover, “Broadcast channels,” IEEE Trans. Inf. Theory, vol. IT-18, no. 1, pp. 2–14, Jan. 1972. [6] P. Delsarte, “An algebraic approach to the association schemes of coding theory,” Philips Res. Rep. Suppl., no. 10, pp. 1–97, 1973. [7] L. A. Dunning and W. E. Robbins, “Optimal encoding of linear block codes for enequal error protection,” Inf. Control, vol. 37, pp. 150–177, 1978. [8] E. K. Englund, “Nonlinear unequal error protection codes are sometimes better than linear ones,” IEEE Trans. Inf. Theory, vol. 37, no. 9, pp. 1418–1420, Sep. 1991. [9] E. K. Englund and A. I. Hansson, “Constructive codes with unequal error protection,” IEEE Trans. Inf. Theory, vol. 43, no. 3, pp. 715–721, Mar. 1997. [10] W. J. van Gils, “Two topics on linear unequal error protection codes: Bounds and their lengths and cyclic code classes,” IEEE Trans. Inf. Theory, vol. IT-29, no. 9, pp. 866–876, Sep. 1983. [11] G. L. Katsman, “Bounds on volume of linear codes with unequal information-symbol protection,” Probl. Peredach. Inf., vol. 16, no. 2, pp. 25–32, Apr.–Jun. 1980. [12] G. O. H. Katona, “The Hamming sphere has minimum boundary,” Stud. Sci. Math. Hungarica, vol. 10, pp. 131–140, 1975. [13] V. I. Levenshtein, “Upper-bound estimates for fixed-weight codes,” Probl. Peredach. Inf., vol. 7, no. 4, pp. 3–12, Oct.–Dec. 1971. [14] M. C. Lin and S. Lin, “Codes with multi-level error correcting capabilities,” Discrete Math., vol. 83, pp. 301–314, 1990. [15] S. Litsyn, “New upper bounds on error exponents,” IEEE Trans. Inf. Theory, vol. 45, no. 3, pp. 385–398, Mar. 1999. [16] B. Masnick and J. K. Wolf, “On linear unequal error protection codes,” IEEE Trans. Inf. Theory, vol. IT-3, no. 10, pp. 600–607, Oct. 1967. [17] R. J. McEliece, E. R. Rodemich, H. Rumsey, Jr, and L. R. Welch, “New upper bounds on the rate of a code via the Delsarte–MacWilliams inequalities,” IEEE Trans. Inf. Theory, vol. IT-23, no. 3, pp. 157–166, Mar. 1977. [18] R. H. Morelos-Zaragoza and S. Lin, “On a class of optimal nonbinary linear unequal error protection codes for two sets of messages,” IEEE Trans. Inf. Theory, vol. 40, no. 1, pp. 196–200, Jan. 1994. [19] F.Özbudak and H. Stichtenoth, “Constructing linear unequal error protection codes from algebraic curves,” IEEE Trans. Inf. Theory, vol. 49, no. 6, pp. 1523–1527, Jun. 2003. [20] A. Samorodnitsky, “On the optimum of Delsarte’s linear program,” J. Combin. Theory, ser. A, vol. 96, no. 2, pp. 261–287, 2001.
Yogesh Sankarasubramaniam, Student Member, IEEE, and Steven W. McLaughlin, Fellow, IEEE
Abstract—This correspondence presents two variable-rate encoding al) constraint when = 2 +1, gorithms that achieve capacity for the ( +1 is not prime. The first algorithm, symbol sliding, is a genor when eralized version of the bit flipping algorithm introduced by Aviran et al. In addition to achieving capacity for ( 2 +1) constraints, it comes close to capacity in other cases. The second algorithm is based on interleaving and is a generalized version of the bit stuffing algorithm introduced by Bender and Wolf. This method uses fewer than biased bit streams to achieve ) constraints with + 1 not prime. In particular, capacity for ( +2 1) constraints 2 requires the encoder for ( only biased bit streams. )-constrained sequences,
Index Terms—Bit flipping, bit stuffing, ( Shannon capacity.
I. INTRODUCTION A binary sequence is said to be (d; k)-constrained if successive “1”s are separated by at least d and at most k consecutive “0”s. A (d; k) code is defined as an invertible mapping between unconstrained binary sequences and (d; k)-constrained sequences. There is a long history of (d; k ) codes, and they are part of virtually all magnetic and optical disc recording systems today (see [4] for an overview). The d constraint is used to regulate intersymbol interference and the k constraint is important for timing recovery. Let us denote by R(d; k), the encoding rate of a (d; k) code. It is well known that the maximum possible value of R(d; k) is equal to the Shannon capacity of the (d; k) constraint given by [8]
C (d; k) = log2 d;k ; where d;k is the positive, real root of the characteristic equation Gd;k (z ) = 1, and Gd;k (z ) is the characteristic polynomial of the (d; k ) constraint, given by
Gd;k (z ) =
k+1
j =d+1
z 0j
z 01 + z 0(d+1)
when k < 1 when k = 1:
(1)
R(d;k) measures how close the The encoding efficiency E (d; k) = C (d;k) code is to capacity. A (d; k) code is said to be optimal if it is 100% efficient. In this correspondence, we present optimal code constructions for two classes of (d; k) constraints. The proposed codes are variable-rate, in the sense that for any given finite input block length, the generated output block length can vary depending on the actual input bits. In [6] and [7], optimal (d; k) codes have been studied from a source coding perspective. This can be understood as follows. For given nonnegative integers d and k; 0 d < k , let Xd;k = f0k 1; 0k01 1; . . . ; 0d+1 1; 0d 1g if k < 1, and let
Manuscript received June 21, 2004; revised February 17, 2006. This work was supported by Seagate Research. The material in this correspondence was presented in part at the International Symposium on Information Theory and Its Applications, Parma, Italy, October 2004. The authors are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: yogi@ece. gatech.edu; [email protected]). Communicated by K. A. S. Abdel-Ghaffar, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2006.876224 0018-9448/$20.00 © 2006 IEEE
