EUROCODE 1990: EUROCODE '90 pp 94-105

# Unidirectional error detecting codes

• Gérard D. Cohen
• Luisa Gargano
• Ugo Vaccaro
Section 2 Combinatorial Codes
Part of the Lecture Notes in Computer Science book series (LNCS, volume 514)

## Abstract

A fixed-length binary code is called t-unidirectional error detecting if no codeword can be transformed into another codeword by at most t unidirectional errors. In this paper we consider the problem of mapping information sequences of length k into code-words of a t-unidirectional error detecting code of length k+p. In case of systematic codes we show that the parameters p and i must satisfy the relation t≤2p −2p/2+1+p. Moreover, we give a simple systematic encoding to map information sequences into codewords of a t-unidirectional error detecting code. In case of non-systematic codes, we give a method to design t-unidirectional error detecting codes in which the number p of check bits must satisfy the inequality t≤2 p p−1. The encoding and decoding algorithms require time linear in the number k of information bits.

## References

1. [1]
S. Al-Bassan and B. Bose, “On Balanced Codes”, IEEE Int. Symp. Inform. Theory, Japan, June 1988.Google Scholar
2. [2]
S. Al-Bassan and B. Bose “Design of Efficient Balanced Codes”, 19th Int. Symp. on Fault-Tolerant Computing, Chicago, Ill., 1989.Google Scholar
3. [3]
J. M. Berger, “A Note on Error Detecting Codes for Asymmetric Channel”, Information and Control, vol. 4, pp. 68–73, 1961.
4. [4]
J. M. Borden, “Optimal Asymmetric Error Detecting Codes”, Information and Control, vol. 53, pp. 66–73, 1982.
5. [5]
B. Bose, “On Unordered Codes”, 17th Int. Symp. on Fault-Tolerant Computing, Pittsburgh, Penn., pp. 102–107, 1987.Google Scholar
6. [6]
B. Bose and D. J. Lin, “Systematic Unidirectional Error-Detecting Codes”, IEEE Trans. Comp., vol. C-34, pp. 1026–1032, 1985.Google Scholar
7. [7]
R. M. Capocelli, L. Gargano, G. Landi and U. Vaccaro, “Improved Balanced Encodings”, IEEE International Symposium on Information Theory, San Diego, USA, 1990.Google Scholar
8. [8]
R. M. Capocelli, L. Gargano and U. Vaccaro, “An Efficient Algorithm to Test Immutability of Variable Length Codes”, IEEE Trans. Inform. Theory, vol. IT-35, pp. 1310–1314, 1989.
9. [9]
R. M. Capocelli, L. Gargano and U. Vaccaro, “Efficient q-ary Immutable Codes”, Discrete Applied Math., to appear.Google Scholar
10. [10]
R. W. Cook, W. H. Sisson, T. G. Stoney, and W. N. Toy, “Design of Self-Checking Microprogram Control”, IEEE Trans. Comput., vol. C-22, pp. 255–262, 1973.Google Scholar
11. [11]
C. V. Freiman, “Optimal Error Detecting Codes for Completely Asymmetric Binary Channels”, Information and Control, vol. 5, pp. 64–71, 1962.
12. [12]
P. Godlewski and G. D. Cohen “Some Cryptographic Aspects of Womcodes”, Proceedings of CRYPTO '85, Lecture Notes in Computer Science, Springer Verlag, 1985.Google Scholar
13. [13]
J. R. Griggs, “Saturated Chains of Subsets and a Random Walk”, Journal of Combinatorial Theory, Series A, vol. 47, pp. 262–283, 1988.
14. [14]
D. E. Knuth, The Art of Computer Programming, Vol. 1, pp. 70 and 486, Addison-Wesley, Reading, Mass., 1968.Google Scholar
15. [15]
D. E. Knuth, “Efficient Balanced Codes”, IEEE Trans. Inform. Theory, vol. IT-32, pp. 51–53, 1986.
16. [16]
E. L. Leiss, “Data Integrity on Digital Optical Discs”, IEEE Trans. Computers, vol. C-33, pp. 818–827, 1984.Google Scholar
17. [17]
E. L. Leiss, “On Codes which are Unchangeable under Given Subversions”, J. Combin. Inform. & Syst. Sci., vol. 10, pp. 91–109, 1985.Google Scholar
18. [18]
E. L. Leiss, “On Testing for Immutability of Codes”, IEEE Trans. Inform. Theory, vol. IT-33, pp. 934–938, 1987.