Backstory: Lester Saunders Hill wrote unpublished notes, about 40 pages long, probably in the mid- to late 1920s.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 fourth section of his paper. I hope to post more later. (Part 3 is here.)
Section 4: Index of the reference matrix
There will be considerable advantage in employing a reference matrix of index .
Consider the sequence which is to be transmitted.
Errors may be made in transmittal, and the sequence may come through in the form
where , , where , , the errors and being, of course, elements of the finite field . If (resp., ) is correctly transmitted then the error (resp., ) vanishes, and (resp., ).
Let us consider what happens when out sequence is mutilated to the extent of errors, all of which affect the (the being supposed correctly transmitted).
There is no real loss of generality if we assume that the errors affect the first of the
, the errors being , . Since the have been correctly transmitted, the mutilated message can not be in check unless
But we recall that, by definition, . Hence the message will fail to check, and the presence of error will be disclosed, unless the errors satisfy the system of equations
for , of which the determinant is
But our matrix is supposed to be of index , so that $\Delta \not=0$. It follows, by Lemma 1, that the system (S) has no solution other than , for . Hence errors, confined to the , can not escape disclosure if the reference matrix is of index . By the same argument, applied a fortiori, a set of less than errors, confined to the , will certainly be disclosed when is of index .
If the reference matrix is of index , we can make the following statement:
Whenever the sequence is transmitted with errors not affecting the , and not more than in number, the presence of error will infallibly be disclosed.
We shall hereafter understand that is of index .