A check digit system over a group of arbitrary order

2013 8th International Conference on Communications an Networking in China (CHINACOM) A check igit system over a group of arbitrary orer Yanling Chen Chair of Communication Systems Ruhr University Bochum D-44780 Bochum, Germany Email: yanlingchen-q5g@rube Markku Niemenmaa Department of Mathematical Sciences University of Oulu 90014 Oulu, Finlan Email: markkuniemenmaa@oulufi A J Han Vinck Institute for Experimental Mathematics University of Duisburg-Essen D-45326 Essen, Germany Email: vinck@iemuni-uee Abstract In this paper, we stuy a check igit system over an abelian group of an arbitrary orer The work is a generalization of a recently introuce check igit system which is base on the use of elementary abelian p-group of orer p k Furthermore, we challenge the current stanars by comparing our system with several well-known an wiely use systems such as ISBN, MEID, ISAN an the system over alphanumeric characters We show that our esign outperforms all of them in terms of the error etection capability I INTRODUCTION Accoring to the statistical investigations by D F Beckley [1] an J Verhoeff [2], when transmitting a sequence of igits, the most common transmission errors mae by human operators are the following: 1) single error: a b ; 2) ajacent transposition: ab ba ; 3) twin error: aa bb ; 4) jump transposition: abc cba ; 5) jump twin error: aca bcb For their relative frequency of occurrence, one can refer to Table I Note that insertion an eletion errors are not inclue in above list since they can be etecte easily if all coewors transmitte are of equal length The recognition of these errors is esirable an attracts talente researchers to esign systems with a satisfying error etection capability To o so, it is usually one by appening a check igit a n+1 to a given sequence a 1 a n of information igits (The use of two check igits is not avise since stuies have shown that the absolute number of errors that occur roughly oubles when the number of igits increases by two, as pointe in [4]) Accoring to Table I, it is clear that 1) single errors an 2) ajacent transpositions are the most prevalent ones So research attention was first brought to esign systems over groups with anti-symmetric mappings which ensure these two kins of errors to be etecte One can refer to a long list of research articles such as [3], [4], [5], [6], [7], [8] an a survey of anti-symmetric mappings in ifferent groups in [9] In aition, possibility of constructing error etecting coes base on quasigroups was iscusse in [5] Necessary an sufficient conitions were establishe in orer to etect 2) ajacent transpositions an 4) jump transpositions (but only in the information igits) The control igit involving in both errors was taken into account in [10] A comprehensive TABLE I ERROR TYPES AND THEIR FREQUENCIES [9] Frequency in % Error type Description in symbol Verhoeff Beckley single error a b 790 86 transposition ab ba 102 8 jump transposition acb bca 08 twin error aa bb 06 phonetic error (a 2) a0 1a 05 6 jump twin error aca bcb 03 other error 86 investigation on a check igit system over a quasigroup was conucte in [11], [12], where necessary an sufficient conitions were establishe in orer to etect each of the 5 error types So far, the approaches have been taken are in general mathematically analytical In [13], M Niemenmaa propose a check igit system for hexaecimal numbers, base on a suitable automorphism of the elementary abelian group of orer 16 Its esign is concise an elegant, with the capability of etecting all the 5 types of errors Inspire by this simple but effective esign, the authors of [14] propose check igit systems over a group of orer p k, for a prime p an k 1 Their systems coul achieve the same error etection capability an beyon, over groups of prime power orer which is a generalization of hexaecimal numbers In this paper, we further exten their results by proposing check igit system over a group of an arbitrary orer The iea is to employ several parallel subsystems as introuce in [14] which are base on the use of elementary abelian p-groups of orer p k It is worth mentioning that in [14] the authors looke into the following two categories of jump errors, which inclue an further exten the error types 2)-5) 6) t-jump transposition: 7) t-jump twin error: ab 1 b t c cb 1 b t a ab 1 b t a cb 1 b t c It is easy to see that error types 2) ajacent transposition an 4) jump transposition, can be regare as t-jump transpositions for t = 0 an t = 1, respectively; 3) twin error an 5) jump twin error, can be regare as t-jump twin errors for t = 0 an t = 1, respectively 897 978-1-4799-1406-7 2013 IEEE

These two kins of errors were first consiere in [15] an treate as transposition an twin errors on places (i, i+t+1), where 1 i n an i + t + 1 n They are of interest, not only because they simplify the list of the error types, but also because they may occur more frequently than expecte, especially when people input ata while using a new keyboar with an unexpecte layout, or when they forget to switch the language to the right one they inten to use In real life applications, there are many well-known examples for use of check igit systems, such as the International Stanar Book Number (ISBN) Coe, the European Article Number (EAN) Coe, the Universal Prouct Coe (UPC), the International Stanar Auiovisual Number (ISAN), the International Mobile Equipment Ientifier (MEID), The rest of the paper is organize as follows: First in Section II, we introuce some preliminaries In Section III, we briefly review the recently introuce check igit system over a group of a prime power orer In Section IV, we present our general esign of a check igit system over a group of an arbitrary orer In Section V, we challenge current stanars by comparing our system with several well-known an wiely use systems such as ISBN, MEID, ISAN an the system over alphanumeric systems Finally we conclue in Section VI II PRELIMINARIES For a given check igit system, we enote t to be the longest jump length such that for any t t, all the t- jump transpositions an t-jump twin errors will be etecte Intuitively we have the etection raius: R = t +1, reflecting the capability of the system to etect these two kins of generalize jump errors By efinition, a system capable of etecting error types 1)-5) has an etection raius R 2 Let t c be the maximum t that coul be achieve, accoringly R c = t c + 1 be the longest etection raius Then a check igit system capable of etecting all the single errors an ouble errors of types 6) an 7) within R c, is of interest ue to its esirable error etection capability In [14], the authors propose systems over a group of orer p k, an emonstrate that { 2 k 2 p = 2; R c = (1) p k 1 2 1 p o prime They also provie easy construction of such systems that etect all the single errors an achieve the maximum etection raius R c We will briefly review their constructive approach in Section III Note that in case of k = 1, accoring to (1), we have R c = 0 for p = 3, an R c = 1 for p = 5, respectively This implies that their propose systems over a group of orer 3 an 5, respectively, are unable to etect all the errors of types 1)-5) This is consistent with the note as state in [15, Remark 5] III CHECK DIGIT SYSTEM OVER A GROUP OF ORDER p k In this section, we review the esign of the check igit system propose in [14] which is base on the use of elementary abelian p-groups of orer p k Their propose system has the ability to etect all the five error types 1)-5) an beyon A Group Consier a check igit system over a set of p k numbers: 0, 1, 2,, p k 1, where p is a prime an k > 0 One can represent these p k numbers as elements of the abelian group G = Z p Z p Z p (2) }{{} k Each number correspons to a k-tuple in G For simplicity, one can take the k-tuple to be its base p representation For instance, consier the hexaecimal numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E an F One can represent them as elements of the abelian group G = Z 2 Z 2 Z 2 Z 2 by enoting 0 = (0, 0, 0, 0), 1 = (0, 0, 0, 1),, 9 = (1, 0, 0, 1), A = (1, 0, 1, 0), B = (1, 0, 1, 1), C = (1, 1, 0, 0), D = (1, 1, 0, 1), E = (1, 1, 1, 0), an F = (1, 1, 1, 1) B Check equation Suppose that the information igits a 1,, a n an the check igit a n+1 are all interprete as elements of the group G Then a n+1 can be etermine by the check equation a 1 P + a 2 P 2 + + a n P n + a n+1 P n+1 = 0, (3) where P is a k k matrix over the prime fiel F p Note that the operations in the check equation are conucte in F p C Check igit system As one can see in (3), P is the only an the key parameter which etermines the performance of the system in terms of error etection In particular, we recall the following theorem: Theorem 31: [14, Theorem 44 & Theorem 46] Let P be a k k matrix whose characteristic polynomial is a primitive polynomial over F p Then a check igit system over an abelian group of orer p k built on P by using the check equation (3) is able to etect all the single errors an ouble errors of type 6) an 7) within etection raius R c as efine in (1) It is known that there are φ(p k 1)/k primitive polynomials of egree k over F p, where φ( ) is Euler s Totient function As inicate in [14], given any of the primitive polynomials, an easy construction of matrix P suitable for Theorem 31, is to take its companion matrix Recall that the companion matrix of a monic polynomial g(z) = c 0 + c 1 z + + c k 1 z k 1 + z k of positive egree k over F p is efine to be the following k k matrix 0 0 0 0 c 0 1 0 0 0 c 1 0 1 0 0 c 2 0 0 0 1 c k 1 The character polynomial of such a matrix is exactly g(z) 898

IV CHECK DIGIT SYSTEM OVER A GROUP OF ORDER p k1 1 pk2 2 pk In this section, we exten the esign of the check igit system over a group of orer p k to a general group of arbitrary orer N > 1 Let N = p k1 1 pk2 2 pk be the unique prime factor ecomposition of N We assume that p 1, p 2,, p are istinct primes in the orer such that p k1 1 < pk2 2 < < pk A Group First we recall the funamental theorem of finite abelian groups that every finite abelian aitive group of orer N can be expresse as a irect sum of elementary abelian p j - groups of orer p kj j, where 1 j So we can express the abelian group G of orer N to be the irect sum of G j, ie, G = G 1 G 2 G, where G j is the elementary abelian p j -group of orer p kj j as escribe in Section III-A: ie, for 1 j, G j = Z pj Z pj Z pj }{{} (4) k j Consier a check igit system over a set of N numbers Note that the N numbers, from 0 to p k1 1 pk2 2 pk 1, can be represente as elements of the abelian group G Each number x correspons to a -tuple in G For simplicity, we choose the representation (x 1, x 2,, x ) such that x = q 1 p k1 1 + x 1; q 1 = q 2 p k2 2 + x 2; q 2 = q 3 p k3 3 + x 3; q 2 = q 1 p k 1 1 q 1 = x Note that for x {0, 1, l=j+1 p k l l 1 j ; an j=1 pkj + x 1; (5) j 1}, the quotient q j < for 1 j 1; the remainer 0 x j < p kj j 1 x = x 1 + x 2 p k1 1 + x 3 p k1 1 pk2 2 + + x p k l l (6) Note that x j can be further inteprete as an element in G j an in the sequel we will use its base p j representation in the calculation Therefore, given information igits a 1,, a n, corresponingly we have their representations in G ie, for 1 i n, l=1 for a i (a i,1, a i,2,, a i, ) G (7) Furthermore, for 1 j, a i,j can be regare as an element in G j, ie, a i,j (a i,j (1), a i,j (2),, a i,j (k j )) G j (8) where a i,j (l) Z pj for 1 l k j B Check equation Let the information igits a 1,, a n an the check igit a n+1 be interprete as elements of the group G, ie, a 1 (a 1,1, a 1,2,, a 1, ) a 2 (a 2,1, a 2,2,, a 2, ) a n (a n,1, a n,2,, a n, ) a n+1 (a n+1,1, a n+1,2,, a n+1, ) Consier the j-th coorinate of a 1,, a n an a n+1, for 1 j We have a 1,j, a 2,j,, a n,j an a n+1,j, all of which are elements of G j Thus similar to the approach as escribe in Section III, one can etermine a n+1,j from a 1,j, a 2,j,, a n,j by applying the following check equation: a 1,j P j +a 2,j P 2 j + +a n,j P n j +a n+1,j P n+1 j = 0, (10) where P j is a k j k j matrix in the prime fiel F pj, an the operations in (10) are conucte in F pj In a parallel manner, one can employing subsystems over G 1, G 2,, G, respectively, to obtain (a n+1,1, a n+1,2,, a n+1, ) which correspons to a n+1 accoring to (6) C Check igit system We note that the representation efine in (5) is a bijection So when a transmission error occurs, they will be reflecte by the representations of the erroneous igits at at least one of the coorinates Recall that for each coorinate j, there is an unerneath check igit system which is over G j with a etection raius R j, where 1 j It is easy to see that the overall system over G coul achieve a etection raius R no less than min{r 1, R 2,, R } Therefore, we easily have the following theorem as a generalization of Theorem 31 Theorem 41: For 1 j, let P j be a k j k j matrix whose characteristic polynomial is a primitive polynomial over Z pj Then a check igit system over a group of orer p k1 1 pk2 2 pk built on {P j, 1 j } by using the check equation (10) is able to etect all the single errors an ouble errors of type 6) an 7) within the etection raius R = 2 k1 2 p 1 = 2 & 2p k1 1 < pk2 2 ; (p k2 2 1)/2 1 p 1 = 2 & 2p k1 1 > pk2 2 ; (11) (p k1 1 1)/2 1 otherwise Here p k1 1 < pk2 2 < < pk is assume Remark: Although we use a bijection as efine in (5) to represent numbers in {0,, j=1 pkj j 1} as elements of the abelian group G, however it is worth mentioning that such a bijection is not unique Another alternative is to represent any x {0,, j=1 pkj j 1} to be a -tuple (x 1, x 2,, x ), where x j = x mo p kj j for 1 j Given (x 1, x 2,, x ), one can retrieve x by applying the Chinese Remainer Theorem (9) 899

V CHALLENGING CURRENT STANDARDS In this section, we compare our system with some well known an willy use systems such as ISBN, MEID, ISAN an so on A Check igit system over Z 11 Example 51: The International Stanar Book Number coe (ISBN) is over Z 11, an uses the check equation: 10 i x i 0 mo 11 (12) This system etect all the errors of types 1)-5) with the exception of the twin error at places (5, 6) as 5 + 6 = 11 In [15, Example 18], a moification of the ISBN coe is propose over Z 11, which uses the following check equation: 5 10 i x i + (5 j) x j 0 mo 11 (13) j=6 This system improves the ISBN coe in the manner that it etects all the errors of types 1)-5) without exceptions In our approach, we apply Theorem 41 for p = 11 an k = 1 Note that over Z 11, the primitive elements are 2, 6, 7, 8 We use α {2, 6, 7, 8} in the following check equation: 10 α i x i 0 mo 11 (14) For instance, if we take α = 6, Equation (14) has coefficients {α i mo 11, 1 i 10} = {6, 3, 7, 9, 10, 5, 8, 4, 2, 1} In particular, the MOD 11-2 system specifie in ISO/IEC 7064 [17] can be consiere as a special case of our system by taking α = 2 By Theorem 41, our system coul etect not only the errors of types 1)-5), but also more generalize errors of types 6)- 7) within etection raius R c = 4 It is easy to check that this hols also for the moification system efine by (13) In fact, both systems are able to etect all the single errors, all the t-jump transposition errors, an almost all the t-jump twin errors with the only exception of the 4-jump twin error The reason is that in our system, α 5 10 mo 11 an thus α i + α i+5 = 11 α i, for 1 i 5; whilst in the moification system propose in [15, Example 18], the coefficients at places (i, i + 5) are i an i, respectively, which sum to 0 As a irect result, the 4-jump twin errors remain unetecte in both systems Moreover, our system outperforms the original ISBN coe an the moification system on etecting the phonetic errors ( a0 1a, where a 2) It only fails to etect the phonetic errors as x i = 10 for 1 i 9 However, in the ISBN coe, number 10 is not use for x i, 1 i 9 at all So we coul say that our system is able to etect all the possible phonetic errors occurre in ISBN; whilst both the original ISBN coe an the moification propose in [15, Example 18] fail to o so (one can refer [15, Example 18] for a etaile list of unetectable phonetic errors for both systems) B Check igit system over hexaecimal numbers Example 52: A Mobile Equipment IDentifier (MEID) [18] is a globally unique 14-igit hexaecimal ientification number for a physical piece of mobile station equipment They are use as a means to facilitate mobile equipment ientification an tracking an therefore shoul be resistant to moification An MEID is compose mainly of two basic components, the manufacturer coe an the serial number, as shown in Table II For an MEID which contains at least one hexaecimal igit in the RR igits, the check igit is calculate using a slight moification of the Luhn formula, in the manner that all arithmetic is performe in base 16 One can refer to [19, Annex B] for the calculation Note that the check igit is not part of the MEID an is not transmitte when the MEID is transmitte It is easy to check that MEID as a hexaecimal check igit system, coul not etect the following errors: t-jump transposition error: (F, 0) (0, F) at places (i, i + t) for all o t such that 1 i 14, i + t 14 t-jump twin error: (x, x) (x + 5, x + 5), where 3 x 7, at places (i, i + t) for all o t such that 1 i 14, i + t 14 Example 53: The International Stanar Auiovisual Number (ISAN) [20] is a numbering system that enables the unique an persistent ientification of any auiovisual works An ISAN consists of 16 hexaecimal igits, which can be ivie into two segments: root segment an episoe segment, as shown in Table III An appene check igit is calculate over the 16 ISAN igits accoring to a MOD 37, 36 system specifie in accorance with ISO/IEC 7064 [17] However, accoring to [17], the MOD 37, 36 system is unable to etect all the errors of types 1)-5) In fact, it fails to etect about 016% of error type 2), 28% of error types 3) an 5), an 17% of error type 4) So both MEID an ISAN, two wiely use hexaecimal check igit systems, fail to etect all the errors of types 1)-5) Example 54: Following our approach as escribe in Theorem 31, one can construct alternative systems for MEID an ISAN as follows: Represent the hexaecimal numbers as elements of the abelian group G = Z 2 Z 2 Z 2 Z 2 as escrible in Section III-A Fin a 4 4 matrix P whose characteristic polynomial is either x 4 + z + 1 or z 4 + z 3 + 1 (both are primitive polynomial of egree 4) For instance, we choose the companion matrix of x 4 + z 3 + 1, 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 as an easy choice of such P Apply the following check equation to calculate the check 900

TABLE II THE FORMAT OF MEID MEID Manufacturer Coe Serial Number Check Digit R R X X X X X X Z Z Z Z Z Z C x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 TABLE III THE FORMAT OF ISAN ISAN Root Episoe Check Digit R R R R R R R R R R R R E E E E C x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 x 15 x 16 x 17 igit a n+1 from the n information igits a 1,, a n : n+1 a i P i = 0 It is easy to see that the above system serves as an alternative for MEID by letting n = 14, an an alternative for ISAN by letting n = 16 Comparing to both MEID an ISAN, our system is capable of etecting all the errors of types 1)-5) Beyon that, it is able to etect both the t-jump transposition an t-jump twin errors within its etection raius R c = 14 In more etail, our system, as an alternative for MEID, is capable of etecting all the possibles errors of types 1) an 6)- 7); whilst as an alternative for ISAN, is capable of etecting all the single errors, an almost all the possible jump errors of types 6)-7) with the only exception of 14-jump transposition an twin errors at places (1, 16) an (2, 17) C Check igit system over hexatriecimal numbers Alphanumeric is a combination of alphabetic an numeric characters, containing in total 36 symbols Since it is quite popular in human interfaces, it is of great interest to evelop check igit system over hexatriecimal numbers A MOD 37, 36 system as specifie in accorance with ISO/IEC 7064 [17], belongs to such a category an use in the real life application for livestock ientification Example 55: Following our approach as escribe in Theorem 41, one can construct alternative alphanumeric systems as follows: Since 36 = 2 2 3 2, we can represent the hexatriecimal numbers 0 to 35 as elements of the abelian group G = G 1 G 2 where G 1 = Z 2 Z 2 an G 2 = Z 3 Z 3 Given n characters, a 1,, a n, we easily have (q 1,, q n ) an (r 1,, r n ), where q i an r i are the quotient an remainer, respectively, when iviing a i by 9 for 1 i n Note that the base 2 representation of q i an base 3 representation of r i are actually use in calculation To set up the check equation, we nee two matrices: one binary matrix P 1 an the other ternary matrix P 2, such that their characteristic polynomials are primitive polynomials over the prime fiels F 2 an F 3, respectively That is, P 1 shoul be a matrix which has x 2 +x+1 (which is the only primitive polynomial of egree 2 in F 2 ) as its characteristic polynomial; an P 2 shoul be a matrix whose characteristic matrix is one of the 3 polynomials: x 2 + 2x + 2, x 2 + 1, an x 2 + x + 2 (which are the only 3 monic primitive polynomial of egree 2 in F 3 ) For instance, we can choose ( ) ( ) 0 1 0 1 P 1 = ; P 2 = 1 1 1 2 From the n information igits a 1,, a n, we have (q 1,, q n ) an (r 1,, r n ) Apply the following check equations to obtain (q n+1, r n+1 ) : n+1 n+1 q i P i 1 = 0 over F 2 ; r i P i 2 = 0 over F 3 The check igit a n+1 = q n+1 9 + r n+1 By Theorem 41, the above system has a etection raius R = 2 an thus can etect all the most frequent error types 1)-5) It performs better than existing alphanumeric systems in ISO/IEC 7064 such as MOD 37, 36 system (with one check igit) Recall the fact that the MOD 37, 36 system fail to etect all the most frequent 5 types errors In fact, accoring to [17], the MOD 37, 36 system fails to etect about 016% of error type 2), 28% of error types 3) an 5), an 17% of error type 4) There are another two alphanumeric systems state in ISO/IEC 7064: MOD 37-2 an MOD 1271-36 However, MOD 37-2 is over 37 characters ( 0 to 9, an A to Z, plus * ) an MOD 1271-36 is with 2 check igits VI CONCLUSION In this paper, we stuy a check igit system over an abelian group of an arbitrary orer The work is a generalization of a recently introuce check igit system which is base on the use of elementary abelian p-group of orer p k Furthermore, we challenge the current stanars by comparing our system with several well-known an wiely use check igit systems such as ISBN, MEID, ISAN an the system over alphanumeric characters We show that our esign outperforms all of them in terms of the error etection capability In aition, our approach is in general simple, constructive an easy to be aopte in practice an it therefore serves as an attractive alternative for ISBN coe, MEID, ISAN, alphanumeric ientification system an for other applications 901

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