FOR LARGE SERIAL NUMBERS. R. C. Bose. University of North Carolina. Institute of Statistics Mimeo Series No March 1965

Size: px

Start display at page:

Download "FOR LARGE SERIAL NUMBERS. R. C. Bose. University of North Carolina. Institute of Statistics Mimeo Series No March 1965"

Chloe McDowell
5 years ago
Views:

1 .. e ERROR DETECTNG AND ERROR CORRECTNG ndexng SYSTEM> FOR LARGE SERAL NUMBERS R. C. Bose University of North Carolina nstitute of Statistics Mimeo Series No. 426 March 1965 Given a set of serial numbers, they can be indexed by adding a few redundant digits, calculated according to certain rules. f X is the original serial number, then the number X* obtained after addition of the redundant digits is called the index of X. The purpose of adding the redundant digits is to enable us either to detect, or to both detect and correct the more common errors which are liable to be made in reporting X*. The most common errors are (i) wrongly reporting a single digit (error of type ) (ii) reversal of two adjacent digits (error of type ). n this paper we describe two indexing systems. n the first system X is a six digit number, the digits being 0,1,2,3,4,5,6,7,8,9. Each X is indexed by adding no more than one redundant digit. Errors of type and in reporting X* can be detected. n the second system we consider an octal numerical system based on the 8 digits 0,1,, 7. We take X to be a seven digit serial number. t is indexed by adding three suitably chosen redundant digits. Errors of type and in reporting X* can be correc~ea. This research was supported by the National Science Foundation Grant No. GP-3792 and the Air Force Office of Scientific Research Grant No. AF-AFOSR DEPARTMENT OF STATSTCS UNVERSTY OF NORTH CAROLNA Chapel Hill, N. C.

2 .. e Error Detecting and Error Correcting ndexing Systems for Large Serial Numbers R. C. Bose University of North Carolina 1. ntroduction. Given a set of serial numbers, they can be indexed by adding a few redundant digits, calculated according to certain rules. X is the original serial number, then the number X* obtained after addition of the redundant digits is called the index of X. f The purpose of adding the redundant digits is to enable us either to detect, or to both detect and correct the JOOre common errors which are liable to be made in reporting X*. An indexing system is useful when information has to be stored with respect to a large number of individuals. a la.'rge clientele. he quotes in any correspondence. For example consider a firm with An index number is to be assigned to each client whicl1 The clients card on 'tmch the information relative to him is stored is identified by means of this index number. No..., the index number may be wrongly reported. The most collloon errors are (i) wrongly quoting a single digit (error of Type ), (ii) reversal of t...ro adjacent digits (error of type ). t would be useful to have an indexing system which is self-checking with respect to common errors, 1.e., When a client makes such an error it is readily detected. Also it would be desirable to have a system in which When an error has been detected, it should be possible to correct it. This research was supported by the National Science Foundation Grant No. GP-3792 and the Air Force Office of Scientific Research Grant No. AF-AFOSR-76o-65.

3 n this paper we describe two indexing systems. n the first system X is a six digit number, the digits being 0,1,2,3,4,5,6,7,8,9. Each X is indexed by adding no nx:>re than one redundant digit. Errors of type and in reporting x* can be detected. n the second system vie consider an octal numerical system based on the 8 digits 0, 1,, 7. We take X to be a seven digit serial number. redundant digits. t is indexed by adding three suitably chosen Errors of type and n in reporting X* can be corrected. 2. An Error Detecting ndexing Syste~ Suppose we have a set of six dibit serial numbers (the digits being 0,1,2,3,4,5,6,7,8,9). We give here an example of an indexing system employing no mare than one redundant digit and able to detect either a single error in one digit (including the redundant digit) or an error consisting of a reversal of two consecutive digits. These two types of error we shall call errors of type and type respectively. We shall also show that if an error has been detected then (under the assumption that the error belongs to one of the two types described above) there are only a few possibilities among which the true serial numbers must lie. Let the serial number to be indexed be X = x6~x4~~:lcj. where x. (i = 1,2,3,4,5,6) is one of the digits 0,1,, 9. Then there J. is a unique integer x such that o (2.1) X o == -(~+X4+x6) + (xl+~+x5) (nx:>d 11), S X o S 10 f < x < 9, we say that the index of X is the seven digit number ' ' ' el

4 .. e f X o = 10, we say that the index of X is the same as X. Thus in this case x* = X, is a six digit number. For example if X = , then x == -7 (rod 11). Hence x = and X is indexed by X* = Again if X = , x == -l(rod 11). 0 Hence x = 10, and X is indexed by X* = o N~l suppose we allow the possibility that in reporting x* one of the two types of errorsconsidered has been committed. a+l c == (-1) (ya - x ex), 3 (mod 11). Let the reported value be y* = Y6Y5Y4Y3Y2Y1YO in case, when X* is a seven digit number and y* = Y6Y5Y4Y3Y2Yl in case, when X* is a six digit number. We can find a unique integer c such that 0 ~ c ~ 10, and where Yo is taken to be 10 in case. Then if the index has been correctly reported c = O. Suppose now that in reporting X* an error of type has been cornrnit- ted, then there is an integer a such that Y a ~ X a whereas y. = x. ~ ~ for i ~ a. n case, 0 ~ a ~ 6 and i = 0,1,2,, 6. n case, 1 < a < 6 and i = 1,2,,6. Then from (2.1) and (2.2) (2.3) Since 0 <Y a - ~~ 9, c ~ o. Thus we will conclude that an error has been committed. Again suppose that in reporting X* an error of the second type has been committed. Then there is an integer a such that Y a = x a +1 ' Y a + l = x a Whereas Yi = Xi for i~ a or a + 1. Also vl'c can assume that X a ~ X a + l ' otherwise there would be no error. n case, 0 ~ a ~ 5 and

5 i = 0,1,2,, 6. n case, 1 ~ a ~ 5 and i = 0,1,2,, 6. Then from (2.1) and (2.2) we have, (2.1~) Since 0 < been committed. what error has been committed. Example 1. a+l c =(-1) 2(X a + - X l a ), (mod 11). ~+l- xal ~ 9, c" O. Thus we will conclude tnt an error has Notice that the non-vanishing of the check number c does not tell us x = then X* = For the purpose of illustration let us suppose that as 5. Then y* = Now 0 ~ c ~ 10 n reporting X* suppose 8 is wrongly reported Hence c = 8, and we conclude that an error has been committed. We can reconstruct the possible values of X* from y* using the fact that c = 8, and assuming that the error is of type or type, i.e., there is an error in a single digit or there has been a reversal of two digits. of type has been committed in reporting x a then from (2.3), We x a = Y a + (_l)a c, (mod 11). can make the following table: Table 1 Y a x a Correct x* '( f an error ell ' c =-( ) + ( ), (mod 11).

6 .. e then have Again it is possible that the discrepancy is due to a reversal of two consecutive digits in reporting X*. y* Table 2 ~ 5P:S:i:l: :* x Example 2. Let us next suppose that X = c =- ( ) + (8+7+9) 3. An error correctillg indexing system. n this case we can reconstruct X* by reversing two consecutive digits of y* and check whether (2.1) holds x x x x Since x must be the same as the last digit in X* the only possio bility is that the correct X* was Thus altogether there are 8 different values of X* which could have lead to Y*. Consider an octal system in which we use only the eight digits 5 x 0 (mod 11). Then x o X* = ; ; ; ; ; We = 10 and X* = Suppose there is an error of type reversing the position of the digits 2 and 7. Then X* would be reported as y* = Now o ~ c ~ 10, and Hence c = 10, which shows that an error has been conunitted. As in example 1, we find that any of the following 6 values of X* could have lead to y*: 0,1,2,3,4,5,6,7. A seven digit number in this system will take care of 8 7 = 221 = 2,097,152 or roughly two million serial numbers. We shall now

7 show that by adding 3 redundant digits, suitably calculated, one can correct any single error in a digit (error of type ), or any error consisting of a reversal of two adjacent digits (error of type ). Consider the Galois field GF 8 of 8 e1enents. Let the elements be 0: 0 = 0,0:1,0:2,ex;,0:4,0:5,0:6'~. t is well known [1,2, ] that the non-zero elements can be written in t'fq different forms: (i) The multiplicative form where each non-zero element is a power of a primitive element t (ii) The additive form where each element is a quadric polynomial of t with coefficients 0 or 0: 0 = 0 = 0 = to ~ = 1 0: 2 = t = t ex; = t 2 = t 2 C\ = t 3 = 1 + t 2 0: 5 = t 4 = 1 + t + t 2 0: 6 = t 5 = 1 + t 0: 7 = t 6 = t + t 2 To add two elements we take the additive form and remember that the coefficients of the polynomials add (mod 2). Thus ex; + ~ = t 2 + (1+t+t 2 ) = 1 + t = 0: 6 To multiply two elements we take the multiplicative form and remember that t 7 = t o Thus 6 _ Element Multiplicative form Additive form

8 .. e We can identify the digits 0,1,2,3,4,5,6,7 ~dth the elements ao,a1,a2'~'04,a5,a6,a7 respectively Then any seven digit octal number X can be made to correspond with a 7-vector! with elements from GF S. Consider the vector! = (x 1,x 2, 000, ~), where the Xi's belong to GFso Let X be encoded to so that x8,x9'~0 where H is the matrix given by!* = (X1'~'..., ~, xs ' x 9 ' x 10 ), are determined by the relation x:*'ht=o, t 0 0 H = t t 2 t 3 t 4 t 5 t This gives the encoding rules: t 2 xs = t 6 (x 1 +~+~+x4 +x5+x6+~) x 9 = t ~+ t2~ + t3~ + t4x4 + t 5 x 5 + ta.x 6 + t 7 x 7 x 10 = t 5 (x 2 + x4 + x6) Then the index of X is X* the octal number corresponding to the vector 2f* lor example to find the index of 2,401,546 vte identify it with the vector (a2'04,ao,a1'~'04,a6) or (t, t 3, 0, 1, t 4, t 3, t 5 ) :. Xs = t 7+ t 2 + t 6 + t 3 + t 2 + t 4 = 1 + t 2 + (t+t 2 ) + (1+t 2 ) + t 2 + (1+t+t 2 ) 7

9 x = t2 + t S + t 4 + t 2 + t S 9 = t 2 + (1+t) + (1+t+t 2 ) + t 2 + t 2 + (l+t) = 1 + t = t S = 0: 6 x = t 8 + t S + t 8 10 = t + (1+t) + t = 1 + t = 0: 6 :.!* = (0:2,C\,0:0,,\,~,C\,0:6,C\,0:6,0:6) Hence the index of X = 2,~01,546 is x* = 2,401,546,466. We shall now show that if an error of type or type is committed in reporting X* it can be corrected. Let X* be reported as y*. Let x* and 'if.!' be the vectors (with elements from GF 8 corresponding to X* and y* respectively. The error vector e* is defined by ~* = Y:.* - x*. f an error of the type has been committed, so that the i-th digit of X* has been wrongly reported, then e* = (0,0,, Yi-xi' 0) Now Y:.* is known to us. We may therefore calculate the syndrome of t* which may be defined as Y:.* i! = (!* + e*)jix = e* H! = e.h i ' J.-,.,here e i = y. - Xi and h. is the i-th row vector of vector of H T J. -J. 8 _

10 .. e are identical. Again suppose there is an error of type in reporting x* so that the i-th and {i+l)-th digits have been reversed. Since the field GF (1 Let S = e* = y* - x*" Nov; the syndrome of y* is y*rr = (x*" + e*)h T = e~ rr () o = (o,o,, xi+1-x i, xi-x i + 1,, 0) is of characteristic 2, addition and subtraction to 1 t Now it should be noted that the last three columns of S are 9 o o Then = z.(h. + h. 1), where h. and h'+ l are the i-th and {i+l)-th row vectors of rrr. -~ -~ The sum of i-th and (i+l)-th columns of H is given by the i-th col~~ of S, where proportional to 1 1 o 0 t is now easy to check that no two of the 19 columns of H and S are proportional to one another. f there is one error in a digit or a single reversal error, the syndrome of y* "lill be proportional to just one row of H T or st. n the first case we conclude that there is an error of type, and in the second case we conclude that there is an error of type.

11 ~ne position of the column will give the position of the error and the constant of proportionality will give e. or z. as the case may be. can then be corrected. ~ ~ The error E;KBJ.Ple. We shall novt illustrate the procedure of error correction. We have seen that the index of X = 2,401,546 is X* = 2,401,546,466. Suppose that there is an error of type, so that the digit 4 in the 6-th position in X* has been reported as 2, then y* = 2,401,526,466. Hence where Hence tl1e syndrome of ~ is N01.:r ~ = (t,t3,0,1,t4,t,t5,t3,t5,t5), ~ H T = (a 1,a 2,a.,), a 1 = t + t t 4 + t + t 5 + t = t 4, a 2 = t 2 + t t 4 + t t t = t 3, a., = 0 + t t t 7 = t 4 (t 4, t 3, t 4 ) = t 4 (1, t 6, 1) (1, t 6, 1) is the 6-th row of rt. Hence we conclude that there is an error of type in the 6th digit, and e6 = t 4 Now ~ =t. Hence ~ = Y"l + e6 = t + t 4 = t 3 = 0: 4 Hence we correct the 6-th digit of y* to 4, getting back the correct X* = 2,401,546,466. Now suppose there has been an error of type n, say the 5-th and 6-th digits of X* have been reversed giving y* = 2,401,456,466. ~ = (t, t 3, 0, 1, t 3, t 4, t 5, t 3, t 5,t 5 ), ~HT = (b 1, b 2, b 3 ), 10 Then _ :,

12 .. e where b 1 = t + t t 3 + t 4 + t 5 + t = 0, b = t 2 + t t 4 + t + t 3 + t t = t 4 2 b 3 = 0 + t t t 7 = t Hence the syndrome of ;[* is (0, t 4, t) = t(o, t 3, 1) He note that 0, t 3, 1 is the 5-th row of ST. Hence we conclude an that there has been/interchange in the 5th and 6th digits. Hence intercl~ing the 5-th and 6-th digits in y* we get back the correct which check~. x* = 2,401,546,466 The value of z5 is t. This should agree with x 5 -x 6 But t 4 3 X 5 - x6 (};5 - Cl4 = - t = t, References 1. R. C. Bose. On the application of the properties of Galois fields to the problem of construction of hyper-graeco-latin squares. Sankhya 3 (1938), pp R. D. Carmichael. ntroduction to the theory of groups of finite order. New York, Dover publications (1956), Chapter X 11

Institute of Statistics Mimeo Series No April 1965

Institute of Statistics Mimeo Series No April 1965 _ - ON THE CONSTRUCTW OF DFFERENCE SETS AND THER USE N THE SEARCH FOR ORTHOGONAL ratn SQUARES AND ERROR CORRECTnm CODES by. M. Chakravarti University of North Carolina nstitute of Statistics Mimeo Series