An Iterative Approach for the Correction of Iterative Errors

COORDINATED SCIEN CE LABORATORY

A N ITERATIVE APPROACH FOR THE

CORRECTION OF ITERATIVE ERRORS

ROBERT T. CHIEN S SE JUNE HONG

UNIVERSITY OF ILLINOIS - URBANA, ILLINOIS

AN ITERATIVE APPROACH FOR THE CORRECTION OF ITERATIVE ERRORS

by

Robert T. Chien Se June Hong

Coordinated Science Laboratory University of Illinois

Urbana, Illinois

This work was supported in part by the Joint Services Electronics Program (Uo S» Army, U.S. Navy, and U.S. Air Force) under Contract DAAB-07-67-C-0199, and in part by the National Science Foundation under Grant No. GK-2339.

Reproduction in whole or in part is permitted for any purpose of the United States Government.

This document has been approved for public release and sale; its distribution is unlimited.

Other CSL Reports in Information Science include:

1» Kasami, Tadao, "An Efficient Recognition and Syntax-Analysis Algorithm for Context-Free Languages," March, 1966, R-257.

2» Barrows, J. T., Jr,, "A Topological Technique for Analysis of Active Networks," August, 1965, R-266.

3. Barrows, J, T., J r „, "A New Method for Constructing Multiple Error Correcting Linear Residue Codes," January, 1966, R-277.

4, Lum, Vincent, "A Theorem on the Minimum Distance of BCH Codes over GF(q)," March, 1966, R-281.

5. Preparata, Franco P„, "Convolutional Transformations of Binary Sequences: Boolean Functions and Their Resynchronizing Properties, March, 1966, R-283o

6. Kasami, Tadao, "Weight Distribution Formula for Some Class of Cyclic Codes," April, 1966, R-285»

7= Kasami, Tadao, "A Note on Computing Time for Recognition of Languages Generated by Linear Grammars," April, 1966, R-287.

8o Lum, Vincent, "On Bose-Chaudhuri-Hocquenghem Codes over GF(q)," July, 1966, R-306,

9. Bahl, Lalit Rai, "Matrix Switches and Error Correcting Codes from Block Designs," August, 1966, Thesis, R-314«

10« Kasami, Tadao, "Weight Distributions of Bose-Chaudhuri-Hocquenghem Codes," August, 1966, Thesis, R-317.

H o Preparata, F - P., Metze, G., and Chien, R. T., "On the Connection Assignment Problem of Diagnosable Systems," October, 1966, R-322.

12 » Chien, R„ T. and Preparata, F. P , , "Topological Structures of Information Retrieval Systems," October, 1966, R-325o

13o Heller, James Ernest, "Decoding Procedures for Convolutional Codes," November, 1966, R-327»

14. Lum, Vincent and Chien, R= T,, "On the Minimum Distance of Bose- Chaudhur i-Hocquenghem Codes," November, 1966,

R-328-/

CSL Reports (continued)

Hsu, Hsung Tsao, "Error Correcting Codes for Compound Channels," December,

1966, R-331o

16o Hong, Se June, "On Minimum Distance of Multiple Error Correcting Arithmetic Codes," January, 1967, R-336.

Gaddess, Terry G., "A Study of an Error Detecting Parallel Adder," January, 1967, M.S. Thesis, R-337.

18o Preparata, Franco P., "Binary Sequence Convolutional Mapping: The Channel Capacity of a Non-Feedback Decoding Scheme, March, 1967, R-345.

19. _{Preparata, F. P. and Chien, R. T., "On Clustering Techniques of} Citation Graphs," May, 1967, R-349.

Lipovski, Gerald J . , "Compatibility and Row-Column Minimization of Sequential Machines," May, 1967, R-355.

2 1

.

22. Chow, David K . s "A Geometric Approach to Coding Theory with Application to Information Retrieval," October, 1967, R-368.

23. Tracey, Robert J., "Lattice Coding for Continuous Channels," December, 1967, R-371.

24. Preparata, Franco P., "A Study of Nordstrom-Robinson Optimum Code," Aprii, 1968, R-375.

25. Preparata, Franco P., "A Class of Optimum Nonlinear Double- Error-Correcting Codes," July, 1968, R-389.

26. Kisylia, Andrew Philip, "An Associative Processor for Information Retrieval," August, 1968, R-390.

27. Beach, Edward J., "A Study of a Feedback Time-Sharing System," September, 1968, R-391.

28. Weston, P., and Taylor, S. M., "Cylinders:: A Data Structure Concept Based on Rings," September, 1968, R-393.

CSL Reports (continued)

29» Bahlj Lalit Rai, "Correction of Single and Multiple Bursts of Error," October, 1968, R-397.

3 0

1968, R-398o

31o

R2

1968, R-399o

3 2 o Tzeng, Kenneth Kai Ming, "On Iterative Decoding of BCH Codes and Decoding Beyond the BCH Bound," January, 1969, R-404. 33

o

Co,

So

Manipulating Language," February, 1969, R-407.

34o

R-408o

35o

Jo

3 6 o Hong, SeJune, "On Bounds and Implementation of Arithmetic Codes," October, 1969, R-437»

37» Chien, R. To and Hong, So J», "Error Correction in High Speed Arithmetic," October, 1969, R-438.

3 8 o Chien, R. T. and Hong, S. J., "On a Root-Distance Relation for Arithmetic Codes," October, 1969, R-440.

39o

To,

For copies of these reports please complete this form and send to Professor R, To Chien

Coordinated Science Laboratory University of Illinois

Urbana, Illinois 61801 My name and address is ________________

AN ITERATIVE APPROACH FOR THE CORRECTION OF ITERATIVE ERRORS*

Robert To Chien Se June Hong

Coordinated Science Laboratory University of Illinois

Urbana, Illinois

This work was supported principally by the National Science Foundation under Grant GK-2339 and, in part, by the Joint Services Electronics Program under Contract DAAB-07-67-C-0199»

2

ABSTRACT

The errors most likely to occur in a high-speed multiplier are called the iterative errors. An arithmetic coding technique for the correction of such error patterns is proposed. We present a class of codes and show its error correcting ability. The unique feature of this code is an iterative decoding method.

3

I. INTRODUCTION

The high-speed multiplier schemes such as the one proposed by MacSorley [ l] have been well investigated and implemented in many computers.

In such a multiplier scheme, the multiplier is divided into blocks of two (or more) bits each and each block is multiplied to the multiplicant to form partial products. The partial products are then appropriately shifted and added in a multi-input parallel adder with minimum carry provisions. The expected error pattern is quite different from either the multiple

independent errors or the burst errors. These errors are shown to be iterative in nature and of the following special form. We let m = the length of a block in bits, r = the number of blocks, and let E be a single iterative error.

k r ~1 mi

Definition 1 E = + 2 £ e.2 , where 0 < k < m and e. = 0 or 1 for all i.

i=0 1 L

A large class of arithmetic codes for the correction of such errors has been developed by Chien and Hong [2,3]. It has been shown that this class of codes has an easy implementation scheme and a nearly optimal

rate. We propose a different class of arithmetic codes here, which is based on the concept of an iterative decoding method for the iterative errors.

Arithmetic codes are designed to detect or correct errors in

digital computations. One such error may change many output digits by propa­ gations. Single error correcting codes are summarized in Peterson [4], and multiple independent error correcting codes have been studied by Barrows [5], Mandelbaum [6], Chang and Tsao-Wu [7] and Chien, Hong, and Preparata [8,9].

Burst error correcting arithmetic codes have been investigated by Stein [10], Chien [11], and Mandelbaum [12].

4

Arithmetic codes are of the form AN, where A is a fixed integer called the generator. N is an integer in the interval (0, B-l), and B is the number of code words. If the code length is n, B is the smallest integer such that AB>2 . In the binary case, A is obviously an odd number. The error correcting capability of ordinary AN codes depends on the minimum distance of the code, which in turn depends on the generator A. A corrupted

signal (correct signal plus error) modulo A is called the syndrome of the error which is the same as the error modulo A. Syndrome of an error, usually denoted as S, then leads to the correct decision of the error through the decoding algorithms.

II. DERIVATION OF THE CODE

It follows from the definition that to correct the error one must correctly determine the polarity of the error, the position of the error (k) and the distribution of the erroneous digits, i.e., the set of e ^ s . The class of codes dealt with in this work is for the cases when the number of

tl

blocks, r, is two to the some power, i.e., r = 2 for some t^ > 1. Note that the length of the code is m2 and 2 -1 is now divisible by 2m + 1 for all 0 < i < t^-1.

The Polarity of Error

m2

Let t„ be some integer less than t,. Consider the positive error -1. Clearly

ti tn

, 2 l-l . ,2 °-l . > _

E = E'= +2k £ e.2mi = 2k £ f .2mi mod 2™2 _1

i=0 1 i=0 1

5

t -1 t

where 0 < < 2 for all 0 < i < 2 ^-1. Thus, each f can have at the most (t^-tg)l's in its binary form. If t^-t^ < -|m, the whole residue

t0~ ^

must have less than (2 m) 1's .

Lemma 1 Given tn > t,- — m f- U

1

.m2 0

V

(

1

S = E mod 2 -1 has less than (2 m) l's if and only if the polarity of error is positive.

Proof We must show that when the polarity is negative, S has greater than

t0~1 m2fc0

(2 m) l's. Let E = - E 1 and S' = E 1 mod 2 -1. We know that S' has to~1

less than (2 m) l's. Therefore,

„m2 , ,

S = 2 -1 -S'

0

V

V

and the number of ones in S is greater than m2 - (m2 ) = m2 . W e mention here that S = 0 only if E = 0, i.e., no error.

Q.E.D

Intermediate Error Pattern

Using the same notation as E = + E', or

E ' = 2k I e.2ml i=0 L

(

₂

we now define an intermediate error pattern as

£ . = E ' mod 2m2 ~l-1

6

for all tQ < j < t^. Clearly, £ fc = E , and from Eq. (2) we have

V 2 j'1

£ . = 2k S a.2mi (4)

J i=0 L

t l"j m 2 J

where 0 < a. < 2 and 0 < k < m. Also, note that £. = £ , mod 2 -1

“ 1 “ " J J+l

for all j < t ^ .

Consider an intermediate error pattern, £ ^ , given in an ordinary binary form. Each a^ becomes a burst of length at most t^-j with at least m ~(b-^"j) 0's in between. These bursts can be uniquely recognized if

integer such that tf-j < "2 m, i.e., if j > t^- m. Let k. be the maximum

k i tr j J

2 J a^ < 2 for all i, for the given £^ of Eq. (4). C

Now denote by £ . the following equation which is numerically the same as £

J j Clearly, k . > 0. J “

2J-1

Lemma 2 If t^ > t^- — m, £^ can be uniquely determined from the binary pattern of £„, for all t„ < j < t,-l.

Proof tQ > t1- j m implies j > t ^ | m for all given j's. Thus, the

bursts of a ^ ’s are uniquely recognized for all j. Now mark the position of . ..

[y] th bit after the longest burst and each m th bit positions thereafter,

The term, "burst", denotes a binary pattern beginning and ending with l's A single 1 is considered as a burst of length one. A cyclic connection between m 2 J th bit and the first bit is assumed.

7

.m2'

cyclically around the entire length of 2 bits. These marks fall among the 0's separating the bursts. Let the position of the smallest marked bit be k / , we have

, /2“'-l , . / j

2k E (a.2k 'k )"2mi s £ . mod 2m2 -1

i=0 1 J

/ k - k 7 “1 J

Now, change k until (a/2 ) < 2 for all i, for the first time. By the definition of k ., k' = k - k . and k-k' = k. for Eq. (5). We mention here

k] J J

that any time (a^2 J) becomes an odd number, k^ = 0 and the position of the error, k = (k-k.).

j

Q.E.D. ti-J

Suppose a binary pattern of £ is given and (k-k ) and (a^2 J) 's are all decided according to lemma 2. We let

2^+ -l

l' = 2 (k-k j> S b' 2mi

J+1 i=0 L

(6)

t,-j-l k.

where 0 < b. < 2 and b' + b' _j = (a.2 for all i.

- i - l i+2 i

k 2j+“1 • t -j-k.

Lemma 3 Let £ = 2 E b.2m i . 0 < b. < 2 J for all 0 < i < 2 - 1

J i=0 1 1

t -j-1

Proof From Eq. (4), we know that 0 < b < 2 for all i. Also, from the definition of £., a. = b. + b.,_4 for all 0 < i < 2 J-1. Now, since

i . . J i l, i+2J . — — , *

k i V J k i t,-j

(a^2 ) < 2 , (b^ + b^+

2

J

8

The g-Code

Define a class of integers, 3., as the following, p . is a prime

m2 j J J

factor of 2 +1, such that x = m2^ is the least positive solution for

X

2 +1 = 0 mod P

y

2 j+-l

2 6.2 ' ^ 0 mod 2 + 1 implies E e.2 0 mod 3

i=0 1 i=0 1 j

where e. - 1 or 0 for all i. An equivalent condition is

2J-1

E a .2 ^ 0 mod P .

i=0 1 J

(7)

n_

where |a^ j < 2 for Finding the problem. But one can shows a short list of order at least one.

all i and not all a.'s are 0, l

order of given p seems to be a difficult number theory easily find the order by a computer programming. Table 1

P ’s and the orders. It appears that all the P .'s have J Table 1. P . and o r d e r . m _j 3 1 13 3 2 241 5 1 41 5 2 61681 6 1 241 6 2 673 order m _{j order} 2

2

8

2

9

For a given in and r = 2 \ the 3 -code is defined under the follow­ ing assumptions , i) > tQ > m and tQ > 0; ii) there exist p 's (t^ < j < t^-2) of order at least t^-j+2 and |3 t ^ of order 2. When such 3 j's exist we define the generator of the (3-code as

AP = (21”2 -l)Pt0 P to+i ••• P V 1 (8)

We mention here that this generator divides 2mr-l and therefore resembles the form of the generators for ordinary multiple error correcting arith­ metic codes [5-10].

III» ITERATIVE DECODING

The decoding is done by iteratively determining the intermediate error patterns» We first show how is obtained from given £. and present

j+1 J

the complete decoding algorithm. An example follows for illustration.

Lemma 4 Assume the order of 3 is greater than or equal to t^-j+2,

s 8 . mod (3 . if and only if £ = £' , for all t_ < j < t--l.

J+1 J+1 J y j+1 j+1 o - J - 1

Proof We must show that £ . = £ [ , mod 3 . implies £ , = £ ; , . Now,

--- ---

J+1

J+1

j+1

j+1

i=0

10

but

2mi „ _2m(i+2J) mQd

Thus _{2 J -1}

S f (bi_b±H-2 2 J " (b * ~ b ^_,_o J) 1 2mL = 0 mod 3

i=0 i+2

Since I(b £_b i+2j)| - 2 1 ^ by lemma 3 and | b^ - b[. ?j\ < 2 1 ,

ir k. t,-j t.-j-l tt-i+1

l{(bi'b i+2J)2 J - <b i - b i+ 2 j) 5 l - 2 + 2 < 2 ’ for all j. By

the definition of 3 > J

But

{bi-b l+ 2 j)2 J ' (bi'b i+23) ' 0

(bi+ b i+2j)2 J = a.2 J = (b-+b h 2 j)

Therefore b' = b^2 J for all 0 < i < 2^+ -l

Q.E.D

Theroem 9 The 3-codes, when exist, correct all single iterative errors«

Pro°f Let the initial syndrome be = A^N + E = E mod .

m2 ^ tQ 1

Step 1) If h(S mod 2 -1) < m2 , the polarity is positive, and other­

wise negative« (By lemma 1.) If positive S^ = Sq, and if negative Sf = Ap - SQ „ In either case S^^ = E ; mod A^ .

p = _m2 0

~ L “ b i mod 1 ~l° However, the £ obtained now is in binary

0 c0

pattern. Iteratively follow the next step for t <i<t -2 0— — 1 Step 2)

11

/

Step 3) From the binary £., f ind £ .by lemma 2« Using St = E . mod 3., find

J “ J 1 J J

£ uniquely from £^ by lemma 4.

i'2tl“il

K ^ rn 1

Step 4) Let £ n = 2 £ a „2 „ i) If S, = 0 mod 3 -, then a. = 0

t,-1 „ A i 1 tn- l ’ l

1 i=0 1

or 2 for all i and k = k' „ ii) If S^ f 0 mod 3 fc and a, = 0 or 2 for all i, then k = k' +1. iii) If S ^ 0 mod 3 fc ^ and a^ = 0, 1, or 2 for all i, then k = k"

t,-l Step 5) Let £ = £ 1 = 2 E a .2 tl“ i i=0 L „k-k „mi Let £ 2 tl"1i ' = 2k 2 _1 e'. 2mi h 1=0

(9)

where 0 < e[ < 1 for all i and e*+e , , = a.2k k for all i.

" 1 _ 1 1

By the same arguments as lemma 4, £ # = £ = E if and only if t l tl

£" = £ mod 3. " Q.E »D.

ti

V

Example Let m = 6» Table 1 gives 3 = 241 with order 8 and 32 = 673 with order 2» Let t^ = 2, i.e», r = 8. t^ = 1 satisfies the condition for 3 -code, thus

Ap = (212-1)*241*673

the rate of which is approximately 0.4= Suppose the error is (e^,e^,...,e^)

12

(10110101), k = 3 and of positive polarity. £^ - E* mod 2 -1 becomes the following binary pattern with the marks,

T. ([f]

12

0 1 2 3 4 5 6 7 8 9 10 11 - binary positions

0 0 0 0 1 0 0 0 0 1 1 0 — > _{(11) is the longest burst}

1 first mark jsecond mark - marking

k' = l, (a.2k 'k ) = 8, (a 2k 'k ) = 12 k ^ k (k-k2 ) = 3, (aj2 2 ) = 2, (a22 2 ) = 2 — 4 _{applying lemma 2} & 1 = 23 (2 °2° + 3 °26) —> _{Eq. (5)} 0<b'<2s b'+b' = 2, bj+b' = 3 - for Eq. (6)

& 2 = 23 (l°26 0 + l “26oI+ l “26 °2+2-26 3 ) - by lemma 4 Si = Ei # 0 mod 673, thus k = 3 —> _{by step 4.}

e > ° ; e0+ e 4= e l+ e 5=e2+ e 6=e3=e7=1 - E q . (9) , Step 5 8^ = mod 673 when (eQ,e|,. . .,e^) = (10110101) . Thus decoded.

IV. CONCLUSION

The 3 -codes are based on an interesting decoding method, namely, an iterative decoding for the iterative errors. From the syndrome of an iterative error, intermediate error patterns are iteratively decoded, each time doubling the length of the pattern. Although some searching and matching operations are necessary at each step, the unusual feature of this decoding technique may be desired for some applications.

The 3 -codes of high rate do not seem to exist for small m's. H ow­ ever, for large m, it is very probable that such P 's exist. The rate of the 3 -code is generally less than the rate of the codes described in [3]. Again,

13

The decoder design, a theory or a simple method to find the order of 3 and a proof of existence of 3 -codes for large m are interesting

problems for further research. Also, the iterative decoding concept may find a useful application in the polynomial codes.

V. REFERENCES

[ 1] MacSorley, O.L., "High-Speed Arithmetic in Binary Computers," Proc. of IRE, Vol. 49, No. 1, January, 1961.

[2] Hong, S„J., "Arithmetic Codes for High Speed Multipliers," Ph.D. Thesis, University of Illinois, Urbana, Illinois, July, 1969.

[3] Chien, R.T. and Hong, S.J., "Error Correction in High Speed Arithmetic," Submitted for publication in Information and Control, August, 1969.

[4]

[5] Barrows, J.T., Jr., "A New Method for Constructing Multiple Error Correcting Linear Residue Codes," Report R-277, Coordinated Science Laboratory, University of Illinois, Urbana, Illinois, January, 1966. [6] Mandelbaum, D., "Arithmetic Codes with Large Distance," IEEE Trans,

on Information Theory, Vol. IT-13, No. 2, April, 1967.

[7] Chang, S.H. and Tsao-Wu, N.T., "Discussion on Arithmetic Codes with Large Distance," IEEE PGIT-14, January, 1968.

[8] Chien, R.T., Hong, S.J. and Preparata, F. P., "Some Contribution to the Theory of Arithmetic Codes," Proceedings of the First Annual Hawaii International Conference on Systems Sciences, January, 1968. [9] Chien, R.T., Hong, S.J. and Preparata, F.P., "Some Results in the

Theory of Arithmetic Codes," submitted for publication, Report R-417, Coordinated Science Laboratory, University of Illinois, Urbana,

Illinois, May, 1969.

[10] Stein, J.J., "Prime Residue Error Correcting Codes," IEEE Trans, on Information Theory, Vol. IT-9, July, 1962.

[11] Chien, R.T., "Linear Residue Codes for Burst Error Correction," IEEE Trans, on Information Theory, Vol. IT-10, April, 1964.

[12] Mandelbaum, D., "Arithmetic Error Detecting Codes for Communication Links Involving Computers," IEEE Trans, on Communication Technology, Vol. Com. 13, June, 1965.

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DOCUMENT CONTROL DATA - R & D ( S e c u r i t y c l a s s i f i c a t i o n o f t i t l e , b o d y o r a b s t r a c t a n d i n d e x i n g a n n o t a t i o n m u s t

I O R I G I N A T I N G A C T I V I T Y ( C o r p o r a t e a u t h o r )

University of Illinois

Coordinated Science Laboratory Urbana, Illinois 61801

3 . R E P O R T T I T L E

be e n t e r e d w h e n t h e o v e r a l l r e p o r t i s c l a s s i f i e d )

\ z a . R E P O R T S E C U R I T Y C L A S S I F I C A T I O N 2 b . G R O U P

AN ITERATIVE APPROACH FOR THE CORRECTION OF ITERATIVE ERRORS

4 . d e s c r i p t i v e N O T E S ( T y p e o f r e p o r t a n d i n c l u s i v e d a t e s )

5. a u t h o r( S ) ( F i r s t n a m e , m i d d l e i n i t i a l , l a s t n a m e )

CHIEN, Robert T. & HONG, Se June

6 R E P O R T D A T E November, 1969 7 a . T O T A L N O . O P P A G E S 13 7 b . N O . O F R E F S

12

DAAB-07-67-C-0199; also NSF Grant Gk-2339

b . P R O J E C T N O .

R-443

9 b . O T H E R R E P O R T N O ( S ) ( A n y o t h e r n u m b e r s t h a t m a y b e a s s i g n e d t h i s r e p o r t )

1 0 . D I S T R I B U T I O N S T A T E M E N T

This document has been approved for public release and sale; its distribution is unlimited«,

11 S U P P L E M E N T A R Y N O T E S 1 2 . S P O N S O R I N G M I L I T A R Y A C T I V I T Y

Joint Services Electronics Program thru Uo S. Army Electronics Command Fort Monmouth, New Jersey 07703____

1 3 . A B S T R A C T

The errors most likely to occur in a high-speed multiplier are called the iterative errors» An arithmetic coding technique for correction of such error patterns is proposed. We present a class of codes and show its error correcting ability» The unique feature of this code is an iterative decoding method.

DD ,F°™

,1473

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