<b>Optimal Check Digit Systems Based on Modular Arithmetic</b>

FARIA, L.; PINTO, P.E.D.; PEDROZA, N.

doi:10.5540/tema.2017.018.01.0105

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Articles • TEMA (São Carlos) 18 (1) • Jan-Apr 2017 • https://doi.org/10.5540/tema.2017.018.01.0105 linkcopy

Optimal Check Digit Systems Based on Modular Arithmetic ^†

Authorship SCIMAGO INSTITUTIONS RANKINGS

In this article, we discuss check digits systems based on modular arithmetic, which are used worldwide. Check digits were created to eliminate most errors in data input of computational systems. Though old, a discussion about the optimality of the systems used is not found in the literature. We describe the main existing systems worldwide and highlight those adopted in Brazil. We present the necessary improvements in order to make all systems optimal. We also propose a new optimal system with 3 permutations for systems with modular base 10.

Keywords:
Check digits; error detection; modular arithmetic

Error type	Abrev	Format	Frequency (%)
Single errors	SNG	..a.. → ..b..	79.10
Adjacent transposition	TRA	..ab.. → ..ba..	10.20
Alternate transpostion	TRL	..abc.. → ..cba..	0.80
Adjacent twins	TWA	..aa.. → ..bb..	0.50
Alternate twin	TWL	..aca.. ..bcb..	0.30
Phonetic	PHO	..1a.. → ..0a.., a≥2	0.50
Others	OTH		8.60

System	SNG	TRA	TRL	TWA	TWL	PHO	AVE (%)
10 - 2 weights	0	11.11	100.00	11.11	11.11	0	2.02
10 - 3 weights	0	11.11	11.11	49.23	40.73	0	1.59
10 alphanum. 3 weights	7.81	18.02	18.02	48.76	41.57	0	8.53
10 IBM	0	2.22	100.00	6.67	11.11	12.50	1.16
11 complete	0	0	0	14.29	0	6.25	0.10
11 restrict	1.82	1.82	1.82	15.85	1.82	14.61	1.80
11 CNPJ	0.04	0.14	0.14	0.83	0.03	2.36	0.06

k	3	6	10
n	3	6	10
11	-	-	1 2 4 8 5 10 9 7 3 6
13	1 2 4	1 2 3 6 9 5	1 2 3 4 6 8 12 11 9 5
	3 4 8	1 8 3 6 12 5	1 2 3 4 6 12 5 10 9 7
	5 10 7	2 3 6 9 5 10	1 2 3 4 6 12 11 9 7 5
37	1 2 3	1 2 3 4 5 6	1 2 3 4 5 6 7 8 9 10
	1 2 8	1 2 3 4 5 8	1 2 3 4 5 6 7 8 9 11
	1 2 12	1 2 3 4 5 12	1 2 3 4 5 6 7 8 9 12
97	1 2 3	1 2 3 4 5 6	1 2 3 4 5 6 7 8 9 10
	1 2 5	1 2 3 4 5 9	1 2 3 4 5 6 7 8 9 12
	1 2 12	1 2 3 4 5 14	1 2 3 4 5 6 7 8 9 14

Systems	SNG	TRA	TRL	TWA	TWL	PHO	AVE %
Mod 10 3P optimal	0	2.22	2.22	5.93	5.93	0	0.29
Mod 10 Vehroeff (≤6)	0	2.22	4.44	2.22	4.44	0	0.28
Mod 10 Vehroeff (≥7)	0	2.22	4.44	2.22	4.44	3.57	0.30
Mod 11 restrict	1.82	1.82	1.82	1.82	1.82	1.82	1.66
Prime modular bases	0	0	0	0	0	0	0

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Sociedade Brasileira de Matemática Aplicada e Computacional Rua Maestro João Seppe, nº. 900, 16º. andar - Sala 163 , 13561-120 São Carlos - SP, Tel. / Fax: (55 16) 3412-9752 - São Carlos - SP - Brazil
E-mail: sbmac@sbmac.org.br

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1	0123456789	7	3456710298	13	6752498310
2	9460571382	8	8359472106	14	8310795264
3	3620845197	9	0397568421	15	2931568704
4	1290857436	10	0189723564	16	6582741903
5	8340971265	11	0562184739	17	0541823976
6	8142630579	12	4175908236	18	4189053627

1	σ₁ - σ₂	0 3 6 9 2 8 7 6 5 4
2	σ₂ - σ₃	9 2 3 2 4 0 5 1 8 6
3	σ₃ - σ₁	1 5 1 9 4 2 8 3 7 0
4	σ₁ - σ₂	0 9 8 7 6 2 5 8 1 4
5	σ₂ - σ₃	1 4 9 6 0 4 3 1 8 4
6	σ₁ - σ₃	1 7 5 5 2 2 0 7 3 8
7	σ₁(1) - σ₂(0)	1
8	σ₂(1) - σ₃(0)	7
9	σ₃(1) - σ₁(0)	6

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2	0864279135 2743985160	26	3197502468 3854096271
3	0864279135 3854096271	27	3197502468 4965107382
4	0864279135 4965107382	28	3197502468 5076218493
5	0864279135 5076218493	29	3197502468 6187329504
6	0864279135 6187329504	30	3197502468 7298430615
7	0864279135 7298430615	31	3197502468 8309541726
8	0864279135 8309541726	32	3197502468 9410652837
9	0864279135 9410652837	33	4208613579 4965107382
10	1975380246 2743985160	34	4208613579 5076218493
11	1975380246 3854096271	35	4208613579 6187329504
12	1975380246 4965107382	36	4208613579 7298430615
13	1975380246 5076218493	37	4208613579 8309541726
14	1975380246 6187329504	38	4208613579 9410652837
15	1975380246 7298430615	39	5319724680 6187329504
16	1975380246 8309541726	40	5319724680 7298430615
17	1975380246 9410652837	41	5319724680 8309541726
18	2086491357 2743985160	42	5319724680 9410652837
19	2086491357 3854096271	43	6420835791 7298430615
20	2086491357 4965107382	44	6420835791 8309541726
21	2086491357 5076218493	45	6420835791 9410652837
22	2086491357 6187329504	46	7531946802 8309541726
23	2086491357 7298430615	47	7531946802 9410652837
24	2086491357 8309541726	48	8642057913 9410652837

Optimal Check Digit Systems Based on Modular Arithmetic †

Optimal Check Digit Systems Based on Modular Arithmetic ^†