I’ve finally finished LaTeXing Lester Hill’s manuscript. You can download the pdf here: hill-error-checking-notes-unpublished. Just ask if you want a copy of the latex source.
This post is to simply state Hill’s conclusion. It gives a clue as to why the paper was never published. If I had to guess, I’d say it was because he could not solve the problem he stated in that section.
As far as I know it is still unsolved.
Here is Hill’s Conclusion section:
Further problems connected with checking operations in finite fields will be treated in another paper. Machines may be devised to render almost quite automatic the evaluation of checking elements according to any proposed reference matrix of the general type described in Section 7, whatever the finite field in which the operations are effected. Such machines would enable us to dispense entirely with tables of any sort, and checks could be determined with great speed. But before checking machines could be seriously planned, the following problem — which is one, incidentally, of considerable interest from the standpoint of pure number theory — would require solution:
To construct, for a given finite field with elements, and for the checking therein of -element sequence , a reference matrix
without a vanishing determinant of any order, in which the integer is the greatest possible. Corresponding to any , and any — provided that is less than — there is a definite maximum . This maximum should be ascertained, and the reference matrix therefore constructed.
We are not able to communicate a solution of this general problem.
(signed) Lester S. Hill
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