F can be used for two basic methods: pure error correction

Transcription

1 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 5, MAY Undetected Error Probabilities of Binary Primitive BCH Codes for Both Error Correction and Detection Min-Goo Kim, Member, IEEE, and Jae Hong Lee, Member, IEEE Abstract- In this paper, we investigate the undetected error probabilities for bounded-distance decoding of binary primitive BCH codes when they are used for both error correction and detection on a binary symmetric channel. We show that the undetected error probability of binary linear codes can be quite simplified and quantified if weight distribution of the code is binomial-like. We obtain bounds on the undetected error probability of binary primitive BCH codes by applying the result to the code and show that bounds are quantified by the deviation factor of true weight distribution from the binomial-like weight distribution. I. INTRODUCTION OR DATA communication systems, linear block codes F can be used for two basic methods: pure error correction and pure error detection [l, Ch. 41. Pure error detection incorporated with automatic-repeat-request (ARQ) has been widely used and hybrid ARQ systems have been developed where linear codes are used for both error correction and detection in order to enhance throughput of ARQ systems and to incorporate advantages of forward error correction (FEC) and ARQ. In these systems, good bounds on the undetected error probability is of theoretical and practical interest [2, Ch. 6, Ch It has been shown that undetected error probability can be computed theoretically from the weight distribution of (n,k) binary code by examining its 2k codewords or 2n-k codewords of its dual code. However, the computation becomes practically impossible as n, k, and n - k become large. Therefore, there have been many efforts to extract properties of undetected error probability using the weight enumerators of a code or its dual code, because it solves whether a code is useful for error detection or both error detection and correction. In [31 and [41, an (n,k;q) linear code in pure error detection was investigated and defined as proper code if undetected error probability of the code is upper bounded by q-(n-k)). In addition, some codes satisfying the bound are classified. In [5], more results on the upper bound were investigated for linear codes. In [6], necessary conditions were given on the weight distributions of an (n,k) binary linear code for undetected error probability to be upper bounded by 2-(n--k). Paper approved by T. Aulin, the Editor for Coding and Communications Theory of the IEEE Communications Society. Manuscript received November 17, 1994; revised June 21, 1995 and October 8, M. G. Kim was with the Department of Electronics Engineering, Seoul National University, Seoul, Korea. He is now with the Samsung Electronics Co., Seoul, Korea ( kimminguc3dosa.snu.ac.h). J. H. Lee is with the Department of Electronics Engineering, Seoul National University, Seoul, , Korea. Publisher Item Identifier S (96) /96$ IEEE In this paper, we consider what happens on the undetected error probability if weight distribution of a binary linear code is binomial-like. The undetected error probability for bounded-distance decoding of binary linear codes is easily derived when they are used for both error detection and correction. Let C be a t- errors correcting (n, k, t) binary linear code with the minimum Hamming distance of dmin where t satisfies 0 5 t 5 L(dmIn - 1)/2J. Suppose that the zero codeword is transmitted over a binary symmetric channel (BSC) with crossover probability E and all error patterns of weight of h are equally probable. The undetected error events then occur when received words lie within Hamming distance t from a nonzero codeword. It has been shown that Pud(E) is where &(h) is the decoder error probability which is defined as the ratio of number of words with weight h that lie within distance t from a codeword to number of words with weight k in the whole vector space [2, Ch. 61, [7, Ch. 11, [51. PE(~) is given by where t is the correcting radius of the decoder and Al is the number of codewords of weight 1 [S, Ch. 141, 191, [lo], [ll]. However, (1) is not manageable and it is difficult to verify the properties of Pud (E). In this paper, we will show that &(E) can be quantified by /? and reduces to a manageable form if Al is binomiallike such that Al = /? (7) although n, k, and n - k become large. With a fact that large subclasses of binary primitive BCH codes have approximate binomial-like weight distribution shown by Sidel'nikov et al., we will also derive bounds on PUd(e) for bounded-distance decoding of the code [12], [13]. In Section 11, we derive P%~(E) of binary linear codes with binomial-like weight distribution using the decoder error probability. In Section 111, we apply the results on P%~(E) to binary primitive BCH codes.

2 516 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 44, NO 5, MAY UNDETECTED ERROR PROBABILITY OF BINARY LINEAR CODES WITH BINOMIAL-LIKE WEIGHT DISTRIBUTION In this section, we derive the undetected error probability of binary linear codes by means of the decoder error probability and show that the undetected error probability is simplified if a code has binomial-like weight distribution. In most binary linear codes, weight distributions are unknown except for a small codelength code. However, if weight distribution of a binary linear code is binomial-like in a range of weight h, then PE(h) in (2) becomes a constant which is dependent on the error correcting capability of the code. The following theorem shows this fact. Theoreml: If Al is P(y),h - s h + s and 0 I s I h I n - s, then the decoder error probability PE (h) is for given dmin, then the decoder error probability is given by (1, n-t<h<n where Plde(h) and Pude(h) are given by Pro08 See Appendix A. From Theorem 1 and (l), we obtain the following result: if Al is binomial-like such that Al =,O (y), then PUd(e) reduces to and 0, elsewhere which is very simple and manageable. Furthermore, any bound that quantifies the deviation of the true weight distribution from the binomial distribution yields the bound on P.d ( E) quantified by P. Assume that the true weight distribution Al satisfies Pmin (7 ) I I Pmax (7) (5) where Pmin and Pmax denote the minimum and the maximum of P, respectively, then the undetected error probability is bounded as \ 0, elsewhere. Pro08 See Appendix B. In (9) and (lo), N(Z, h;s) is (10) Pud(t,Pmm) I Pud(~) I Pud(E,Pmax). (6) Thus, a method to provide more precise P can improve accuracy of bound on the undetected error probability. In binary linear codes, it has been shown that binary primitive BCH codes has binomial-like weight distribution [2, Ch. 61. Sidel'nikov et al. derived /3 of binary primitive BCH codes and Fujiwara et al. improved it [12], [13]. The details are given in the next section. In practical consideration, it is more general to assume that Ais are one for 1 = 0,n and Ais are P (7) for dmin n - d,,, [2, Ch. 61, [14, Ch. 71. We introduce a corollary to be useful in deriving Pud(E). Corollary 1 is derived from Theorem 1 with modified Al. Corollary 1: If Al is (0, elsewhere where s + h - 1 and s - h + 1 are positive integers. &de (h) and Pzlde( h) are erroneous terms for AZ = 0 in 1 5 I 5 dmin - 1 and n - d, n - 1, respectively, since Ais are actually zero in these ranges. Plde (h) is equal to Pude (n- h) due to symmetry of (9) and N(1, h; s) [S, Ch. 141, [ll], [15, Ch. 11. By inserting (8) into (l), we obtain &(E) of the code which is very close to (4) if Plde(h) and Pude(h) are small enough that they can be neglected. III. UNDETECTED ERROR PROBABILITY OF BINARY PRIMITIVE BCH CODES In this section, we apply the results in Section I1 to binary primitive BCH codes. The weight distributions of binary BCH codes are unknown except for some very low-rate binary BCH codes. However, for a large subclasses of binary primitive BCH codes, Sidel'nikov et al. have shown that the weights of the code have approximate binomial distribution [2, Ch. 61, [12], [13]. They showed that an (n, k, t) binary primitive BCH code of length n = 2" - 1 and the designed distance

3 2 ~ KIM AND LEE: UNDETECTED ERROR PROBABILITIES OF BINARY PRIMITIVE BCH CODES 571 dmin = 2t + 1, where 2t - 1 < 2r"I21 + 1, has weight distributions given by 1:: 1= < dmin A' 2-mt(;)(l + E(l)), dmin 5 15 Ln/2] (12) Ad, Ln/2] <1< n. In (12), E(1) is E(1) = { E:1, 1 is odd, and 2t Ln/2] 1 is even, and 2t Ln/2] (13) where El denotes an erroneous term of weight resulted from binomial approximation. Upper bound on I El I (or lezz I) was derived in [12] where a subscript of 2i was used instead of 1 here, and it is given by h(zp +?/L 7r 2i-1p=l - 1)[2(t - 1)]-2t where b(z) and C(u,x) are functions given in [12]. A more precise value of E2% is obtained by the improved linear programming method but it is not expressed by a closed form [13]. Therefore, we will use (14) in deriving the undetected error probability hereafter. From (12), we see that the deviation factor p of binary primitive BCH codes is 2-mt(l + E(1)). Let E,, be the maximum of IE(1)I then it yields Pmax = 2-mt(l + Emax) and Pmin = 2-mt( 1 - Emax). Thus, from (4) and (6), estimate of bounds on the undetected error probability are obtained as TABLE I DECODER ERROR PROBABILITY OFTHE (511, 493, 2) BCH CODE Bounds on Pdh) Weight h Lower UDEr Pe(h) x 10-l x IO-' x x 10-l x IO-' x x lo-' x10-l x W x lo-' x IO-' x x lo-' x lo-' x x x x x10-' x10-' x IO-' x10-l x lo-' x lo-' x in-' x io-' etc. etc. etc. 'The approximate decoder error probablllty = TdgO(r) = Xl0-l and the difference between upper and lower bounds on Pud(t) is also limited to 1% of (17). Equation (17) can be simplified further by bounds of the type considered in [l, pp , [2, Ch. 61. By inequality involving binomial coefficients, it is knownthatifp<x,o<p<l,andq= 1-p, then 2 < - 2-nE(X,p), providedp < X (18) i=xn a where E(X,p) is the relative entropy between the binary probability distribution X and p, i.e., E(X,P) = H(P) + (A - P)H'(P) - H(X) (19) = log,(vp) + (1- log2 ((1-4/(1 - P)) (20) where H(z) is the entropy function and H'(x) is dh(z)/dz [2, Ch. 61. Let An = (t + 1) then (17) reduces to and. h(l - )V (16) For (n,k,t) binary primitive BCH codes, it was shown that IE(1)I decreases to zero exponentially as n increases for fixed t, or as 1 approaches Ln/2] for fixed n and t [12], [13]. Thus, E, becomes zero if n is large. In [13], it was shown that some binary primitive BCH codes with large n have 1E(l)I < lop2 and are tabulated for various n and t. For these BCH codes, bounds on the undetected error probability become Pud( )!? 2-mt 2 (:) + s=o (:)ch(l (17) h=t+l Iv. EXAMPLES AND DISCUSSION The decoder error probabilities and the undetected error probabilities for some binary primitive BCH codes are calculated. Table I shows bounds on PE (h) and Plde( h) of (5 l l, 493, 2) BCH code. In Table I, it is shown that upper and lower bounds on PE(~) converge on x 10-1 (= 2-18 E:=, (':I) as h increases. In fact, for h > 12, upper and lower bounds are close to x 10-l. This indicates that the deviation of decoder error probability from 2Tmt E:=, ( y ) becomes negligible in this range of h. Table I1 shows that E, decreases exponentially as h approaches Ln/2] for (51 1, 493, 2) BCH code. The undetected error probabilities of binary primitive (51 1, 493, 2), (511, 484, 3), (1023, 1003, 2), and (1023, 993, 3) BCH codes are calculated. Assume that binary phase shift keying (BPSK) is used for transmission of coded symbols, and a coherent receiver is used for detection on an additive white

4 578 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 5, MAY 1996 TABLE I1 MAXIMUM RELATIVERROR OF THE DECODER ERROR PROBABILITY OF (511, 493, 2) BCH CODE ~~ h 7.8 9,lO ~ E, x10-l xlo-' xlo-' h 13,14 15,16 17,18... E, x ~ O - ~ x ~ O - ~ x1w8.. le-lo 1 E-1 1 Lower bound I 5 1E-06 ' 1E-07 U 1E E E-10 1 E-l 1 1E-12 - Upper bound Lower bound I 1E-13 ' i " ' Fig. 2. Undetected error probabilities of (1023, 1003, 2) BCH code and (1023, 993, 3) BCH code on AWGN channel. Fig. 1. Undetected error probabilities of (511, 493, 2) BCH code and (511, 484, 3) BCH code on AWGN channel. Gaussian noise (AWGN) channel. Fig. 1. shows the undetected error probabilities of (511, 493, 2) BCH code and (511, 484, 3) BCH code. It is shown that the undetected error probability of (511, 493, 2) BCH code is a constant which is close to x 10-1 (= 2-18 E:=, ("I)) for Eb/N, less than 3 db and decreases rapidly as $&/No increases. For a (511, 484, 3) BCH code, the constant is close to x lo-'. It is shown that the undetected error probability of (511, 484, 3) BCH code is less than that of (511, 493, 2) BCH code. Fig. 2 shows the undetected error probabilities of (1023, 1003, 2) and (1023, 993, 3) BCH code where constants are noted as x 10-1 and x 10-1 for (1023,1003,2) BCH code and (1023, 993, 3) BCH code, respectively. Bounds on the undetected error probability of (1023, 1003, 2) BCN code are much tighter than bounds of (511, 493, 2) BCH code since E, decreases as n increases. The undetected error probabilities of (511, 494, 2) BCH code and (1023, 1003, 2) BCH code are calculated when these codes are used for both error detection and correction with various correcting radius r, where 0 5 r, 5 t. Fig. 3 shows that the undetected error probability of (511, 493, 2) BCH code is a constant for Eb/N, less than 3 db whose value are close to x 10-1 (= 2-18 E:=, ('i')), x lop3 (= 2-18 E:=, ('il)), and x lop6 (= 2-18 E!=, (':')) for r, = 2,rc = l,rc = 0, respectively. The constant values decrease as much as r, is reduced since the unde- Fig. 3. Undetected error probabilities of (511, 493, 2) BCH code with correcting-radius bc = 0, T, = 1, T, = 2, on AWGN channel. tectable words decrease in proportional to r,. Fig. 4 shows the undetected error probability of (1023, 1003, 2) BCH code where constant values are close to x 10-1 (= 2-" E:=, ( x (= 2-" Et=, ( 'Osz3), and x (= 2rZ0 co s=o ( ) j for T, = 2, T, = 1, T, = 0, respectively. Difference between upper bound and low bound of (1023, 1003, 2) BCH code is smaller than that of (511, 493, 2) BCH code. Both Figs. 4 and 5 imply that it is effective to use a bounded-distance decoder with smallcorrecting radius for both error correction and detection when a channel is very noisy. v. SUMMARY AND CONCLUSIONS In this paper, we derived the undetected error probability Pud( t) for bounded-distance decoding of binary primitive BCH codes when they are used for both error correction and detection on a binary symmetric channel with crossover

5 KIM AND LEE: UNDETECTED ERROR PROBABILITIES OF BINARY PRIMITIVE BCH CODES E01 rc=2 1 E M rc= 1 M Upper bound Lower bound ( I I I I I I I I I I I I I I I I I 8 Fig. 4. Undetected error probabilities of (1023, 1003, 2) BCH code with correcting-radius T, = 0, rc = 1, r, = 2, on AWGN channel. probability E. We showed that the undetected error probability of binary linear code can be simplified and quantified by the deviation factor p of the true weight distribution from the binomial distribution if weight distribution of the code is binomial-like. We also showed that the undetected error probability is bounded by p as Pud(c,Pmin) 5 Pud(c) 5 PUd(, a,) and a method to provide more precise,8 would improve accuracy of Pud( 6 ). From the results, we showed that, for fixed t and with n = 2-1 and 2t - 1 < 2rm/ 1 + 1, the undetected error probability of (n, k, t) binary primitive BCH codes is close to 2Tmt E:=, (t) 2-nE(X+) when codelength is large. The undetected error probabilities were calculated for some binary primitive BCH codes. APPENDIX A PROOF OF THEOREM 1 Let Nh(s) be the inner most summation in (2) with Al = p ( ). Then, from the substitution of a = (s + h - 1)/2, we have h + ~ - 2 a.( s-a ) S a=o =a(;) a=o =a(;) (:) n! (A.3) a!(n - h - s + Q)!(s - a)!(h - a)! h!(n - h)! 2 a!(h - Q )!(S- a)!(n - h - s + a)! 04.4) where the last step is a general binomial identity. Inserting (A.6) into (2) then yields (3). APPENDIX B PROOF OF CCOROLLARY 1 i) d+t 5 h 5 n-d-t Al since I must be in d 5 I 5 n - d if d+ t 5 h 5 n - d - t. Thus, Theorem 1 yields PE(~) = E:=, (:). ii) t + 15 h 5 d + t + 1, n - d - t h 5 n - t - 1. With an assumption of Al = p (7) for n - 1, Theorem 1 yields PE(~) =,8 E:=, (:). However, Ais are actually zero in h - s 5 I 5 min(h + s,d - 1) and max(h - s,n - d + 1) h + s. Thus, enoneous terms for Al = 0 in these ranges should be corrected. The erroneous terms of decoder error probability for Al = 0 yields Plde(h) and Pude(h), respectively, and they are subtracted from p E:=, (t). iii) n - t 5 h 5 n. PE(h) is one since all-ones codeword covers all error pattems of weight h 2 n - t. ACKNOWLEDGMENT The authors would like to thank Dr. T. Aulin and the anonymous reviewers for their useful suggestions for improving the presentation of the material in this paper. REFERENCES [ 11 W. W. Peterson and E. J. Weldon Jr., Error-Correcting Codes, 2nd ed. Cambridge, MA: MIT Press, [2] S. Lin and D. J. Costello, Jr., Error Control Coding: Fundamentals and Applications. Englewood Cliffs, NJ: Prentice-Hall, [3] S. K. Leung-Yan-Cheong and M. E. Hellman, Concerning a bound on undetected error probability, IEEE Trans. Inform. Theory, vol. IT-22, pp , MA r41 _. S. K. Leung-Yan-Cheong, E. R. Barnes, and D. U. Friedman, On some properties Gf the undetected emor probability of linear codes, IEEE Trans. Inform. Theory, vol. IT-25, pp , Jan [SI J. K. Wolf, A. M. Michelson, and A. H. Levesque, On the probability of undetected error for linear block codes, IEEE Trans. Commun., vol. COM-30, no. 2, pp , Feb [6] P. Perry, Necessary conditions for good error detection, IEEE Trans. Inform. Theory, vol. IT-37, pp , Mar [7] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, [8] R. E. Blahut, Theory and Practice of Error Control Codes. Reading, MA: Addison-Wesley, [9] R. J. Mceliece and L. Swanson, On the decoder error probability for Reed-Solomon codes, IEEE Trans. Inform. Theory, vol. IT-32, no. 5, pp , Sept [lo] K. M. Cheung, More on the decoder error probability for Reed- Solomon codes, IEEE Trans. Inform. Theory, vol. 35, no. 4, _ pp. _ , July M. Huntoon and A. M. Michelson, On the computation of the probability of post-decoding error events for block codes, IEEE Trans. Inform. Theory, vol. IT-23, pp , May [12] T. Kasami, T. Fujiwara, and S. Lin, Approximation of the weight distribution of binary linear codes, IEEE Trans. Inform. Theory, vol. IT-31, no. 6, pp , Nov [I31 T. Fujiwara, T. Takata, T. Kasami, and S. Lin, An approximation to the weight distribution of binary BCH codes with designed distance 9 and 11, IEEE Trans. Inform. Theory. vol. IT-32, no. 5, pp , Sept [I41 A. M. Michelson and A. H. Levesque, Error-Control Techniques for Digital Communication. New York Wiley, [15] C. L. Liu, Introduction to Combinatorial Mathematics. New York McGraw-Hill, 1968.

6 580 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 5, MAY 1996 Min-Goo Kim (M 95) was born in Baekryung island, Korea, on August 20, He received the B.S. degree in electronics engineering from the Younsei University, Seoul, Korea in 1988, and the M.E. and Ph.D. degrees in electronics engineering from the Seoul National University, Seoul, Korea in 1990 and 1996, respectively. He has served as an Assistant Manager of the ASIC Center at the Corporate Technical Operations of the Samsung Electronics Co. Ltd., Suwon, Korea, since His primary research interest is the coding theory, especially the performances analysis of linear codes. His research interests also include digital communication and coding for storage systems. Dr. Kim is currently a member of the KITE. Jae Hong Lee (M 86) received the B.S. and M.S. degrees in electronics engineering from Seoul National University, Seoul, Korea, in 1976 and 1978, respectively. He received the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, MI, in From 1978 to 1981, he was with the Republic of Korea Naval Academy, Jinhae, Korea, as an Instructor and Lieutenant. In 1987, he joined the Department of Electronics Engineering at Seoul National University (SNU), Seoul, Korea where he is currently an Professor. He was a Member of Technical Staff at the AT&T Bell Laboratories, Whipany, NJ, during a sabbatical year of From 1992 to 1994, he served as the Chairman of the Department of Electronics Engineering at SNU. From 1992 to 1994, he served as the Head of Operation and Planning Division of the Institute of New Media and Communications at SNU for which he has also served as the Head of Wireless and Satellite Communications Research Division since His current research interests include communication and coding theory, spread spectrum system, and their applications to wireless communications, intelligent transportation systems, and information storage systems. Dr. Lee is a member of the KITE, KICS, and Tau Beta Pi.