Notes on the penny-weighing problem, lossless symbol coding with nonprimes, etc.

The method of construction of lossless symbol coding matrices for one-error correction is illustrated for the case when the prime symbol order is three, and the application of this matrix to the penny-weighing problem is described. This method is then extended to those cases in which the symbol order is
2^2, 2^3, 2^4, 2^6, 3^2, 3^3, 3^4, 3^5, 5^2, 5^3, 5^4, 7^2, 7^3
, and
p^2
, where
p
is any higher prime. This extension is based on the concept of the master iterating matrix. These matrices are given for the first thirteen cases cited, and their existence is demonstrated for
p^2
. This paper concludes with a short description of Zaremba's condition, and its application to various problems, and more particularly to the hypothetical one-error correcting close-packed code with the symbol order 6.