# Lester Hill’s “The checking of the accuracy …”, part 3

Backstory: Lester Saunders Hill wrote unpublished notes, about 40 pages long, probably in the mid- to late 1920s.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 third section of his paper. I hope to post more later. (Part 2 is here.)

Section 3: Preliminary characterization of checking procedure

Our problem is to provide convenient and practical accuracy checks upon a sequence of $n$ elements $f_1$, $f_2$, $\dots$, $f_n$ in a finite algebraic field $F$.

To fix the ideas, let us assume that we are to employ $q$-element checks $c_1$, $c_2$, $\dots$, $c_q$ upon the sequence $f_1$, $f_2$, $\dots$, $f_n$. The checks are to be determined by means of a fixed reference matrix $Q = \left( \begin{array}{cccc} k_{11} & k_{12} & \dots & k_{1n} \\ k_{21} & k_{22} & \dots & k_{2n} \\ \vdots & & & \vdots \\ k_{q1} & k_{q2} & \dots & k_{qn} \\ \end{array} \right)$
of elements of $F$, the matrix having been suitably constructed according to criteria which will be developed in the following pages. We send, in place of the simple sequence $f_1$, $f_2$, $\dots$, $f_n$, the amplified sequence $f_1, f_2, \dots, f_n, c_1, c_2, \dots, c_q$
consisting of the “operand” sequence and the “checking” sequence. The checking sequence contains $q$ elements of $F$ as follows: $c_j = \sum_{i=1}^n k_{ji}f_i,$

for $j = 1, 2, \dots, q$. Considerations of telegraphic economy dictate the assumption, made throughout the paper, that $q\leq n$.

Before laying down specifications for the reference matrix $Q$, we define a matrix of “index” $q$ as one in which no $q$-rowed determinant vanishes.