J.-G.Zhang

Indexing terms: Error-correction codes, Code-division multiple-access systems, Optical multiple-access interference, Fibre-optic systems

Abstract: In code-division multiple-access (CDMA) systems using all-optical signal processing, the optical multiple-access interference (OMAI) degrades the system performance and can ultimately limit the number of active users. To reduce the effect of OMAI, error-correction codes are used in both asynchronous and synchronous fibre-optic CDMA systems. It is shown that the use of asymmetric error-correction binary block codes can not only effectively reduce the bit-error rate, but also increase the maximum number of active users in a constant-bandwidth network more efficiently than using symmetric error-correction binary block codes. Therefore, this permits implementation of a cost-effective fibre-optic CDMA network.

1 Introduction

Code-division multiple-access (CDMA) techniques have recently been proposed for use in fibre-optic net- works with all-optical signal processing which can pro- vide a very high throughput [l-51. In such optical CDMA (OCDMA) systems, the bit-error-rate (BER) performance is degraded by the optical multiple-access interference (OMAI) which comes from all the other active users [14]. This in turn ultimately limits the number of active users in a given network. In principle, the weight w of an OCDMA address code can be increased to reduce the BER for the fixed number of active users in OCDMA systems using either a prime code [l, 51 or an optical orthogonal code (OOC) of A, = A, = 1 [3, 41, where A, and Ac are auto- and crosscor- relation constraints, respectively. However, the use of a larger w results in higher power losses at OCDMA encoder and decoder. Note that optical power loss also limits the capacity of an all-optical CDMA network, which is a power-limited system rather than a band- width-limited one. Moreover, using a larger w causes a higher system cost, because more optical delay lines are employed in the OCDMA system and optical 1 x w splitterlw x 1 combiner of a higher w are required [l]. 0 IEE, 1997 IEE Proceedings online no. 19971454 Paper received 16th October 1995 The author is with Telecommunications Program, School of Advanced Technologies, Asian Institute of Technology, P.O. Box 4, Klong Luang, Pathumthani 12120, Thailand

To combat the OMAI effectively, error-correction codes (ECCs) can be used in OCDMA systems [6-81. This will permit choice of a lower weight for OCDMA address codes to reduce the complexity and power loss of the OCDMA encoderidecoder. As will be explained below, an error-correction code is used before OCDMA encoding is done at each transmitter and after OCDMA decoding is performed at each receiver. Consequently, the encoders and decoders for an error- correction code operate at a rate which is just higher than the users’ data rate by a factor of llR, where R is the code rate of an ECC. Since the value of 1/R is nor- mally chosen to be 5 or smaller in this paper, the sym- bol rate of ECC-encoded signal can be still a reasonable value (e.g. a few hundred Mbitis) for practi- cal applications, and it is therefore feasible for ECC encodersidecoders to be implemented by using conven- tional electronic circuits to avoid optical power loss completely at this stage. Therefore, the use of error- correction codes permits implementation of cost-effec- tive OCDMA networks.

In this paper, a special type of error-correction codes is introduced, called asymmetric error-correction (AEC) binary block codes, to reduce the BER of OCDMA systems and to increase the maximum number of active users for a given BER (even in a con- stant-bandwidth network) more efficiently than sym- metric error-correction (SEC) binary block codes.

2 systems

An incoherent OCDMA network using optical process- ing is illustrated in Fig. 1, where data messages at the active transmitters using an on-off key are first encoded with their desired OCDMA address code- words and are then distributed to each receiver by an optical star coupler. The maximum number of users is equal to NmrLx, of which N,,, users are allowed to be active for the BER I Interleaving is used to ran- domise the interference pulse patterns which may be introduced by using an error-correction code. Each transmitter is assigned an unique codeword from an OCDMA address code. Identical data rates are assumed for all the users and the same effective average power is assumed at the input of each receiver [4]. Moreover, all the optical sources at transmitters are incoherent so that optical-power signals will incoher- ently add in intensity at an OCDMA decoder. As pointed out in [14], only the OMAI is considered as a source of BER degradation. This is because the effects

Channel model for incoherent optical CDMA

IEE Pro,.-Commun., Vol. 144, No. 5, October 1997 316

Input 4 ECC OCDMA

data , encoder 1- lnterleaver encoder

- \

optical - star

--c coupler - OCDMA

Fig. 1 OCDMA network using a binary block code

of thermal or quantum noise on the BER of OCDMA systems can be reduced by increasing the transmitted power or using optjcal amplifiers while the OMAI results are unaffected.

In OCDMA systems using an error-correction binary block code, each data bit ‘I’ from an interleaver is transformed into the desired destination codeword by using an OCDMA encoder. No light is actually trans- mitted when each data bit ‘0’ is issued by the inter- leaver, and the binary bit ‘0’ might be mistaken for a binary bit ‘1’ if OMAI signals are strong enough to cause a false detection (called 0-error) at the receiver. In contrast, a false detection of the data bit ‘I’ is impossible. This is because, with incoherent optical sig- nal processing, light powers always add up [l, 21. The model of incoherent OCDMA systems using on-off key is thus an asymmetric binary channel (called Z- channel), as shown in Fig. 2. At the receiver, the out- put of an optical correlator consists of the desired sig- nal and OMAI signals, so detection errors may occur. After deinterleaving, an ECC decoder is used for cor- recting t or fewer 0-errors which are assumed to be independent,

1-

0 0 Fig.2 Z-channel

3 asymmetric error cclrrection

There are several strategies to apply ECCs to OCDMA systems. Some may treat the incoherent OCDMA channel as a binary symmetric channel, and use, for example, conventional [n,, k,, d] symmetric error-cor- rection (SEC) binary block codes with minimum dis- tance d for correcting up to t, symmetric errors as encountered in conventional electronic communication systems, where n, is the block length, k, is the informa- tion length, and t, is given by [9]

Binary block coides for symmetric and

Here the symbol 1x1 ‘denotes the integer portion of an real value x. This code can necessarily correct up to t,

IEE Proc.-Commun., Vol. 144, No. 5, October 1997

0-errors [9]. The use of conventional binary block codes in OCDMA systems can improve the BER perform- ance, but it is less efficient. This is because only part of their error-correction capability is utilised on an asym- metric binary channel [lo]. To achieve higher coding efficiency, a special type of binary block code, which is useful for correcting asymmetric errors, is introduced to incoherent OCDMA systems in this paper.

A [no, k,, A] asymmetric error-correction (AEC) binary block code of minimum asymmetric distance A can correct tu or fewer 0-errors [ll], where t, is expressed as

For convenience, let As(n, t ) and A,(n, t ) denote the maximum number of codewords in a code of length n correcting up to t symmetric and asymmetric errors, respectively. Thus, they satisfy [9]

t,=n-1 (2)

As(n, t ) L A, (n, t ) 5 min{(t+ l )A , (n , t ) ;A , (n+t , t ) }

for 15 t 5 n (3) Since, for a binary block code of n and k , the number of necessary codewords is equal to 2k which must be less than or equal to A,(n, t ) or A,(n, t), eqn. 3 states that, for given n and t , a [n, k,, A] AEC code of A = t + 1 may exist with k, > k,, and therefore, can achieve a higher code rate R than an ordinary [n, k,, d] SEC code of d B 2t + 1.

4 systems

The exact performance analysis of the generalised OOC-based OCDMA systems is very difficult, so the bounds on the BER are normally used for OOC sys- tems [3, 41. However, the Gaussian approximation to the density function of the OMAI is usually employed for analysing the BER of prime-code-based OCDMA systems [l , 5, 121, because it is easy to use and to com- pute. This approximation is valid for large values of Nu,, according to the well known central-limit theorem [l, 5, 121. In the following, OCDMA systems with opti- cal linear correlation receivers and OOCs or prime codes are considered.

BER performance of uncoded OCDMA

4. I OOC-based OCDMA systems without ECCs The number of codewords N,,, for a (F, K, 1) optical orthogonal code (OOC) of given length F and weight K can be upper bounded by [2, 31

(4)

317

For convenience of analysis, a perfect optimal OOC (PO-OOC) is defined as the (F, K, 1) OOC having length F

F = NmazK(K - 1) + 1 ( 5 ) for PO-OOCs considered here, N,,, is chosen to be equal to Nm,, i.e. all the transmitters are active in a given OCDMA network.

In this paper, PO-OOCs and the chip-synchronous- interference situation (called case A) are considered for OOC-based OCDMA systems. Note that, when a data bit '1' is transmitted by using an OCDMA-address codeword, it can be always recovered correctly at the receiver using an incoherent optical-correlation decoder, as explained in Section 2. Thus, the error probability when a data bit '1' is sent, i.e. p(0 I l), is equal to zero. The BER derived from case A is an upper bound on the exact BER of OOC-based OCDMA systems [3]. In case A, the probability density function for OMAI signal Il is expressed as [4]

and the BER of uncoded OCDMA systems with OOC and optical linear-correlation receivers is therefore given by

pe = P(llO).P(O) +P(Ol') .P(') = P ( 1 IO).P(O)

where Th is a threshold level which takes K - 1 here, because the performance degradation is caused by the OMAI [ 2 4 ] .

4.2 Prime-code-based OCDMA systems without ECCs Let P be a prime number. The maximum number of users N,,, of a prime code with P is equal to P for asynchronous OCDMA and P2 for synchronous OCDMA [l, 51, respectively. Using a Gaussian distri- bution for OMAI, the BERs of both OCDMA systems can be written using a prime code as [5]

1 / - P / d m x P, = - exp(-z2/2)dz (8)

where r = 1.16 and 1 for asynchronous and synchro- nous OCDMA, respectively.

Jz;; --oo

5 BER performance of coded OCDMA systems

The performance of coded OCDMA systems is evalu- ated with both SEC and AEC binary block codes, respectively. As shown in Fig. 1, after deinterleaving, the coding channel for OCDMA is assumed to be a discrete memoryless, binary channel where the inde- pendent errors occur.

318

5.7 OCDMA systems with symmetric error- correction codes For given block length n, and information length k,, [n,, k,, d,] SEC binary linear block codes with the max- imum 'minimum distance' d, have the best SEC capa- bility among ordinary binary linear block codes, where d, is defined as [ 131

dm = max(d1there exists a [nS, k,, 6] code}

1 I k , I n s

The probability of receiving the codeword of a block code incorrectly is given by

(9)

where p is the error probability for a binary channel with OMAI in coded OCDMA systems, i.e. p = p(1 I 0) = p(0 I 1) = 2P,, and P, are given by eqns. 7 and 8 for OOC and prime code, respectively.

An upper bound on the post-decoding BER of SEC- coded OCDMA systems can be obtained by assuming that j incorrect decoding events produce j + t, post- decoding errors when j > t,. Then the post-decoding BER of SEC-coded OCDMA systems can be expressed as [14, 151

5.2 OCDMA systems with asymmetric error- correction codes Since it is too difficult to know code-weight distribu- tions for general [n,, k,, A] AEC binary block codes, it is possible to derive a simpler, but less tight, bound for AEC-coded OCDMA systems. This can be done by considering 0-errors to occur probably in n, coded bits per AEC codeword. Thus, an upper bound on the post- decoding BER of AEC-coded OCDMA systems can be obtained by making the pessimistic assumption that a pattern o f j (> t,) channel errors will cause the decoded word to differ from the correct word in j + t, positions and a fraction 0' + t,)/n, of the k, information bits to be decoded incorrectly [ 151:

where p = p(1 I 0 ) = 2Pe for AEC codes.

6 Numerical results and discussions

The binary block codes principally considered are those having moderate block length and short information length, because encoders and decoders for such error- correction codes are simple and can be suitable for fast parallel decoding algorithms which will be used in high-speed networks. For convenience, a bandwidth- expansion factor with respect to source data signal is defined as: (i) & f F (or P2) for uncoded OCDMA systems using a perfect optimal OOC (or a prime code); (ii) p, F J R (or P2/R) for coded OCDMA systems using a (Fe, K,, 1) PO- OOC (or a prime code) and an error-correction code of rate R .

IEE Pro,.-Commun., Vol. 144, No. 5, October 1997

For convenience, the term 'equivalent PO-OOC' is introduced. The 'equivalent PO-OOC' for a system using a (F, K, 1) PO-OOC and an ECC of rate R is defined as the PO-OOC systems which, without using ECC, can achieve the same BER performance as the given OCDMA system using ECC. For OOC-based OCDMA systems, a saving-bandwidth coefficient q can be defined as

(13) a

rl = P u , e / P c

where p,,, is the factor pu obtained when using an 'equivalent PO-OOC'.

The larger q, the more convenient is using the ECC. If q < 1, the use of PO-OOC systems without ECC is better than using the ECC in given OOC systems.

Numerical examples which show the benefits of coded OCDMA systems are given in Table 1. The

parameters of AEC and SEC binary block codes are taken from [11, 131, where SEC codes have the maxi- mum d, for given values of n, and k, [13], but for AEC codes the lower bound on the code size is usually con- sidered [ 1 I].

In uncoded OOC-based OCDMA systems, to guar- antee that the BER s over a wide range of Nmax, PO-OOCs with a K 2 10 are usually required due to the phenomenon of 'BER floors', as shown in Fig. 3. Unfortunately, the structure of OCDMA encoders/ decoders is more complex when K becomes larger [l]. Moreover, the cost and power loss of OCDMA encod- ersidecoders increase with K. On the other hand, for a lower K the BER is high. From Table 1, it can be seen that the use of both AEC binary block codes and SEC binary block codes can greatly reduce the effects of

Table 1: Performance improvements of OCDMA systems by using error-correction binary block codes

Uncoded Coded to achieve BER < 1.0 x (see note)

[n,, ka, AI AEC code

n e k, A D c rl n s ks 4" D c rl

[n,, ks, d,,,,,l SEC code M w P,

15 4 1 . 0 9 ~ 16 2 8 1448 0.93 26 2 17 2353 0.46"

29 2 19 2624.5 0.51

5 1 . 0 6 ~ 8 2 4 1204 0.90" 14 2 9 2107 0.64t

18 5 5 1083.6 1.25 20 5 9 1204 1.12

23 7 5 989 1.37 23 7 9 989 1.37

6 7 . 4 9 ~ 6 2 3 1353 l .OOt 8 2 5 1804 0.75

15 8 3 845.6 1.60 16 8 5 902 1.50

23 14 3 740.9 1.46" 23 14 5 740.9 1.46"

7 3.93 x 5 2 2 1577.5 0.86 5 2 3 1577.5 0.86

15 11 2 860.5 1.26" 15 11 3 860.5 1.26"

23 18 2 806.3 1.34" 23 18 3 806.3 1.34"

30 4 1 . 2 9 ~ 16 2 8

5 1 . 4 7 ~ 10 2 5

16 4 5

18 5 5

6 1 . 2 7 ~ 6 2 3

13 6 3

23 12 4

7 8.70 x 5 2 2

15 8 3

23 14 3

2888

3005

2404

2163.6

2703

1952.2

1726.9

3152.5

2364.4

2071.6

0.94" 29 2 19

49 7 23

l . l O t 14 2 9

1.12 19 4 9

1.25 20 5 9

1.00 8 2 5

1.38" 13 5 5

1.9It 23 12 7

0.86" 5 2 3

1.68§ 16 8 5

1.91* 23 14 5

5234.5 0.52"

2527 1.07"

4207 0.64

2854.8 0.95"

2404 1.12"

3604 0.75

2342.6 1.15"

1726.9 1.9It

3152.5 0.86"

2522 1.57§

2071.6 1.91*

50 4 1 . 3 8 ~ 18 2 9 5409 0.83 32 2 21 9616 0.47

55 7 25 4722.1 0.95"

5 1 . 6 5 ~ 1 0 - ~ 10 2 5 5005 l . l O t 14 2 9 7007 0.64

16 4 5 4004 1.12 22 4 11 5505.5 l .OOt

18 5 5 3603.6 1.25" 23 5 11 4604.6 l . l g t

6 1.52 x 6 2 3 4503 1.00 8 2 5 6004 0.75

16 6 4 4002.7 1.65* 17 6 7 4252.8 1.55*

23 12 4 2876.9 1.9It 23 12 7 2876.9 1.9It

7 1 . 1 4 ~ 6 2 3 6303 1.24§ 8 2 5 8404 0.93§

15 8 3 3939.4 1.68* 16 8 5 4202 1.57*

23 14 3 3451.6 1.91* 23 14 5 3451.6 1.91* - Note!: the different marks as signed on the values of r indicate the different ranges of BERs for coded systems. No mark signed denotes 1.00 x s BER < 1.00 x '*' denotes 1.00 x s BER < 3.00 x '$' denotes 1.00 x 10-l2 s BER < 1.00 x

't' denotes 1.00 x 10-l' s BER < 1.00 x '5' denotes BER < 1.00 x

IEE Proc-Commun., Vol. 144, No. 5, October 1997 319

OMAI on the system performance and can guarantee that the BER < lo-' over a wide range of N,,, for low values of K. AEC codes appear to be more suitable for OCDMA applications, because the former can achieve a better q than the latter for given values of k and t ( t > 1). From eqn. 13, one can find that the larger q, the less is bandwidth expansion. Although a higher coding efficiency can be achieved by using those error-correc- tion codes with a higher rate R and larger k, their use can result in a longer time delay and a more complex ECC encodeddecoder. When q > 1, the coded OCDMA systems can accommodate a larger number of users for given values of bandwidth and BER. For example, when K = 5 and N,,, = 30, the use of a [18, 5, 51 AEC code can decrease the BER to 9.0 x 10-lo, whereas (2701, 10, 1) equivalent PO-OOC systems of p,,, = 2701 are required to obtain a BER = 9.14 x 10-lo. If K = 5 and N,,, increases to 37, the BER = 1.2 x

and pc = 2667.6 < p,,, for a [18, 5, 51 AEC code and NmaY = 30.

10-7 -

10-9-

10-11 -

Ili W m

I

0 20 40 60 ao 100 number of active users

Fig.3 BER versus the number of active users N,,, = N,, for uncoded and coded asynchronous CDMA systems using a PO-OOC . . . . . . . . . K = 4, [16, 2 , XI AEC code _ _ _ _ K = 4, [18, 2 , 91 AEC code _ _ _ K = 5 , [lX, 5 , 51 AEC code

K = 6, [6, 2, 31 AEC code ~ uncoded systems

Fig. 3 shows the great reduction of the BER which is caused by OMAI. It is clear that various AEC-coded OCDMA systems of lower K can have the equivalent BER performance of uncoded systems with K = 10. Using AEC codes can also increase N,,, for a given BER without bandwidth expansion compared to uncoded OCDMA systems, as shown in Table 2.

Table 2: Comparison of uncoded and coded asynchro- nous OCDMA systems using OOC for BER 5 2.0 x

Uncoded Coded

K "w Bu K " a x na ka A Pc

10 59 5311 5 70 18 5 5 5043.6

10 30 2701 5 30 18 5 5 2163.6 11 50 5501 6 50 23 12 4 2876.92

Fig. 4 shows the BER performance of asynchronous prime-code OCDMA systems, which is dramatically degraded as N,,, increases. Even when P = 19, an uncoded asynchronous OCDMA network can only support I O out of 19 users to transmit messages simul-

320

taneously with BER = 2.0 x However, one can use, for example, a very simple [4, 2, 21 AEC binary block code to ensure that the BER = 1.2 x for all 19 users to transmit data simultaneously in a given net- work of P = 19 without increasing power loss and cost of OCDMA encoderddecoders, compared with using a larger-weight prime code to achieve the same BER. The use of AEC codes in prime-code OCDMA networks can significantly increase the network capacity which is limited by OMAI, as shown in Fig. 4 and Table 3. From Fig. 4, one can find that [lo, 4, 31 AEC-coded OCDMA systems with P = 11 and @, = 302.5 can achieve the same BER as uncoded systems with P = 19 and pU = 361 for N,,, 5 11, and that the BER of coded systems with P = 13 and p, = 422.5 is very close to that of uncoded systems with P = 23 and pU = 529 for N,,, I 13. Therefore, cost-effective OCDMA networks can be implemented by using AEC codes. In both cases, pc < pU means that using an AEC code can also reduce the bandwidth expansion compared with the uncoded OCDMA scheme. This improvement is more efficient when an AEC code of higher code rate is used.

O r P.5 "

-L

- a - Ili w m - -12 0 m 0 -

-16

-20

-2L 0 10 20 30 LO

number of active users Fi 4 BER versus the number of active users N Cf for uncoded and cozd asynchronous CDMA systems using a prime CO& _ - _ [lo, 4, 31 AEC code _ _ _ [15, X, 31 AEC code ~ uncoded systems

Table 3: Comparison of uncoded and coded asynchro- nous OCDMA systems using prime code for BER 5 2.0 x 10-9

Uncoded Coded

P Nact Pu P Nact na k a A Pc

11 4 121 11 10 6 2 3 363

13 5 169 13 13 10 4 3 422.5

17 8 289 17 15 4 2 2 578

19 10 361 19 19 4 2 2 722

23 14 529 23 23 23 18 2 675.94

The BER of synchronous OCDMA systems using prime codes is illustrated in Fig. 5. The BER perform- ance is dramatically degraded as N,,, increases, espe- cially for a larger N,,,. By using a simple [14, 2, 71 code, the coded synchronous OCDMA systems with P = 7 and @, = 343 can achieve the slightly better BER than uncoded synchronous systems with P = 19 and pU = 361 for Nu,, up to 49 (i.e. N,,, for P = 7). Since p, < pU, the bandwidth can effectively be saved by using

IEE Proc -Commun, Val 144, No 5 October 1997

AEC codes. As shown in Fig. 6, the use of an AEC code can increase the capacity of synchronous prime- code OCDMA networks.

80

70

60

0 20 LO 60 80 100 120 1LO

-

-

-

number of active users Fi .5 BER versus the nwrrber of active users N,,, for uncoded and cojed synchronous CDMA system using a prime code

[14, 2, 71 AEC code _ - _ [18, 2, 91 AEC code __ uncoded systems

719 I

I I 17

I I

I

bandwidth-expansion factor Fig.6 N,,, against bandwidth-expansion factor for BER 5 IO-1o in syn- chronous prime-code CDMA system with a [14, 2, 71 AEC code and with- out AEC codes U- - -W [14, 2, 71 AEC code 0- - -0 uncoded

7 Conclusions

The BER performance of asynchronous and synchro- nous OCDMA systems :IS limited by OMAI. The use of binary block codes in such systems can greatly reduce

the effect of OMAI on system performance, so it allows the use of lower-weight OCDMA address codes to achieve the BER 5 Using error-correction codes can also efficiently increase N,,, limited by OMAI. Compared with the case of uncoded OCDMA systems, it is feasible to use error-correction coding to reduce the complexity of OCDMA encodersldecoders and the optical power loss, which is another practical factor limiting the capacity of all-optical networks. Moreover, AEC binary block codes are more suitable than SEC codes for OCDMA applications, because the former not only ensures that the BER s but also can increase either the maximum number of users or the number of active users in a constant-bandwidth network more efficiently than can SEC binary block codes. Therefore, cost-effective systems can be imple- mented by combining AEC codes with OCDMA.

8 Acknowledgments

The author thanks Prof. G. Picchi of the University of Parma (Italy) for help and Dr. G. Migliorini for sup- port.

9 References

1 PRUCNAL, P.R., SANTORO, M.A., and FAN, T.R.: ‘Spread spectrum fiber-optic local area network using optical processing’, J. Lightwave TechnoL, 1986, LT-4, pp. 541-554

2 CHUNG, F.R.K., SALEHI, J.A., and WEI, V.K.: ‘Optical orthogonal codes: design, analysis, and applications’, ZEEE Trans. Znf: Theory, 1989, 35, pp. 595-604

3 SALEHI, J.A.: ‘Code division multiple-access techniques in opti- cal fiber networks - Part I: Fundamental principles’, ZEEE Trans. Commun., 1989, 37, pp. 824-833 SALEHI, J.A., and BRACKETT, C.A.: ‘Code division multiple- access techniques in optical fiber networks - Part 11: Systems performance analysis’, ZEEE Trans. Commun., 1989, 37, pp. 834- 842 KWONG, W.C., PERRIER, P.A., and PRUCNAL, P.R.: ‘Per- formance comparison of asynchronous and synchronous code- division multiple-access techniques for fiber-optic local area net- works’, ZEEE Trans. Commun., 1991, 39, pp. 1625-1634

6 DALE M., and GAGLIARDI, R.M.: ‘Analysis of fiberoptic code division multiple access’. CSI report 92-06-10, EES Department, University of Southern California, USA, June 1992

7 WU, J.-H., WU, J., and TSAI, C.-N.: ‘Synchronous fibre-optic code division multiple access networks with error control coding’, Electron. Lett., 1992, 28, pp. 2118-2120

8 ZHANG, J.-G., and PICCHI, G.: ‘Forward error-correction codes in incoherent optical fibre CDMA networks’, Electron. Lett., 1993, 29, pp. 1460-1462

9 WEBER, J.H., DE VROEDT, C., and BOEKEE, D.E.: ‘Bounds and constructions for codes correcting unidirectional errors’, IEEE Trans. Znf.’ Theorv. 1989. 35. 00. 797-810

4

5

I O SHIOZAKI, A:: ‘Consiructioi f i r ‘&nary asymmetric error-cor-

11 WEBER, J.H , DE VROEDT, C. , and BOEKEE, D.E.: ‘Bounds recting codes’, ZEEE Trans., 1982, IT-28, pp. 787-789

and constructions for binary codes of length less than 24 and asymmetric distance less than 6’, ZEEE Trans. Znf: Theory, 1988, 34, pp. 1321-1331

12 MATSUNAGE, S., and GAGLIARDI, R.: ‘Digital signaling with code-division multiple-access in optical fiber communica- tions’. Technical report CSI-88-02-03, Department of Electrical Engineering, University of Southern California, 1988

13 VERHOEFF, T.: ‘An updated table of minimum-distance bounds for binary linear codes’, ZEEE Trans., 1987, IT-33, pp. 665-680

14 MICHELSON, A.M., and LEVESQUE, A.H.: ‘Error-control techniques for digital communication’ (John Wiley and Sons, New York, 1985)

15 CLARK, G.C. Jun., and CAIN, J.B.: ‘Error-correction coding for digital communications’ (Plenum Press, New York, 1981)

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