# Lester Hill’s “The checking of the accuracy …”, part 6

Backstory: Lester Saunders Hill wrote unpublished notes, about 40 pages long, probably in the mid- to late 1920s. They were titled “The checking of the accuracy of transmittal of telegraphic communications by means of operations in finite algebraic fields”. In the 1960s this manuscript was given to David Kahn by Hill’s widow. The notes were typewritten, but mathematical symbols, tables, insertions, and some footnotes were often handwritten. The manuscript is being LaTeXed “as we speak”. I thank Chris Christensen, the National Cryptologic Museum, and NSA’s David Kahn Collection, for their help in obtaining these notes. Many thanks also to Rene Stein of the NSA Cryptologic Museum library and David Kahn for permission to publish this transcription. Comments by transcriber will look his this: [This is a comment. – wdj]. I used Sage (www.sagemath.org) to generate the tables in LaTeX.

Here is just the sixth section of his paper. I hope to post more later. (Part 5 is here.)

Section 6: Arbitrary distrbution of errors affecting the $f_i$ and the $c_j$

It is highly desirable, although not strictly necessary, to fashion the reference matrix $Q$ in such a manner that every determinant contained in it, of every order, be different from zero.

Reference matrices of a practical type, and without a vanishing determinant of any order, may be construted in certain finite fields $F$ by very direct methods which can be more easily illustrated than described. It will be expedient at this poin, even at the expense of breaking the continuity of our discussion, to introduce concrete examples of operations in finite fields, and to show how reference matrices be conveniently set up in particular cases. The desirability of contriving that such matrices contain no vanishing determinant can then be brought out more effectively. The examination of arbitrary distributions of errors among the $n+q$ elements of a telegraphic sequence

$f_1, f_2, \dots, f_n, c_1, c_2, \dots, c_q$

can be deferred, therefore, until later.

But, before going into details, we will describe the general form of reference matrix to be employed in all cases.