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On Addition and Multiplication with Hensel Codes
December 1990 (vol. 39 no. 12)
pp. 1417-1423

It has been stated by R.N. Gorgui-Naguib and R.A. King (1986) that the operations of addition and multiplication on Hensel codes originally defined by E.V. Krishnamurthy, T.M. Rao, and K. Subramanian (1975) are seriously in error in that it is possible to add/subtract or multiply Hensel codes and not get a valid Hensel code. It is shown that it is the presence of so-called invalid Farey fractions that results in the need to modify the original arithmetic operations. However, this also results in the Hensel codes becoming redundant. The authors show how to include the invalid Farey fractions such that it is possible to compute with their Hensel codings without the need to map back and forth between the rationals and their Hensel codings. This provides an alternative to the method of Gorgui-Naguib and King. Unfortunately, it turns out that Hensel codes of a large size will be needed in practice, even for relatively small problems.

[1] E. V. Krishnamurthy, T. M. Rao, and K. Subramanian, "p-adic arithmetic procedures for exact matrix computations,"Proc. Indian Acad. Sci., vol. 82A, pp. 165-175, 1975b.
[2] R. T. Gregory and E. V. Krishnamurthy,Methods and Applications of Error-Free Computations. New York: Springer-Verlag, 1984.
[3] R. N. Gorgui-Naguib and R. A. King, "Comments on 'Matrix processors usingp-adic arithmetic for exact linear computations',"IEEE Trans. Comput., vol. C-35, pp. 928-930, Oct. 1986.
[4] N. Koblitz,p-adic Numbers, p-adic Analysis, and Zeta-Functions. New York: Springer-Verlag, 1977.

Index Terms:
Hensel codes; addition; multiplication; Farey fractions; codes; digital arithmetic.
C.J. Zarowski, H.C. Card, "On Addition and Multiplication with Hensel Codes," IEEE Transactions on Computers, vol. 39, no. 12, pp. 1417-1423, Dec. 1990, doi:10.1109/12.61062
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