# Lester Hill’s “The checking of the accuracy …”, part 9

We may inquire into the possibility of undisclosed errors occurring in the transmittal of the sequence:

$\begin{array}{ccccccccccccccc} f_1 & f_2 & f_3 & f_4 & f_5 & f_6 & f_7 & f_8 & f_9 & f_{10} & f_{11} & f_{12} & c_1 & c_2 & c_3 \\ 5 & 17 & 13 & 21 & 0 & 8 & 6 & 0 & 11 & 0 & 11 & 11 & 6 & 15 & 2 \\ X & T & Y & P & V & Z & R & V & H & V & H & H & R & I & F \\ \end{array}$

Invoking the theorem established in sections 4 and 5, and formulated at the close of section 5, we may assert:

• (1) If not more than three errors are made in transmitting the fifteen letters of the sequence, and if the errors made affect the $f_i$ only, the $c_j$ being correctly transmitted, then the presence of error is certain to be disclosed.
• (2) If not more than three errors are made, all told, but at least three of them affect the $f_i$, then the presence of error will enjoy only a

$1-{\rm in}-22^3\ \ \ \ \ (1-{\rm in}-10648)$
chance of escaping disclosure.

These assertions result at once from the theorem referred to. But a closer study of the reference matrix employed in this example permits us to replace them by the following more satisfactory statements:

• (1′)
If errors occur in not more than three of the fifteen elements of the sequence $f_1f_2\dots f_{12}c_1 c_2 c_3$, and if at least one of the particular elements $f_{11}f_{12}c_2$ is correctly transmitted, the presence of error will certainly be disclosed. But if exactly three errors are made, affecting presicely the elements $f_{11}f_{12}c_2$, the presence of error will enjoy a $1$-in-$22^2$ ($1$-in-$484$) chance of escaping disclosure.
• (2′)
If more than three errors are made, then whatever the distribution of errors among the fifteen elements of the sequence, the presence of error will enjoy only a
$1$-in-$22^3$ ($1$-in-$10648$) chance of escaping disclosure.

Assertions of this kind will be carefully established below, when a more important finite field is under consideration. The argument then made will be applicable in the case of any finite field. But it is worthwhile here to look more carefully into the exceptional distribution of errors which is italicized in (1′). This will help us note any weakness that ought to be avoided in the construction of reference matrices.

Suppose that exactly three errors are made, affecting precisely $f_{11}f_{12}c_2$. If the mutilated message is to check up, and the errors to escape disclosure, we must have (for error notations, see sections 4,5):

$11\epsilon_{11}+12\epsilon_{12}=0,\ \ \ 11^2\epsilon_{11}+12^2\epsilon_{12}=\delta_2,\ \ \ 11^3\epsilon_{11}+12^3\epsilon_{12}=0.$
These equations may be written:

$11\epsilon_{11}+12\epsilon_{12}=0,\ \ \ 6\epsilon_{11}+6\epsilon_{12}=\delta_2,\ \ \ 20\epsilon_{11}+3\epsilon_{12}=0.$
But $11\epsilon_{11}=-12\epsilon_{12}$ can be written $11\epsilon_{11}=11\epsilon_{12}$, or $\epsilon_{11}=\epsilon_{12}$.
Etc. In this way, we find that the errors can escape disclosure if and only if

$\epsilon_{11}=\epsilon_{12},\ \ \ {\rm and}\ \ \ \delta_{2}=12\epsilon_{11}.$
The error $\epsilon_{11}$ can be made quite arbitrarily. But then the values of $\epsilon_{12}$ and $\delta_2$ are then completely determined. There is evidently a $1$-in-$484$ chance – and no more – that the errors will fall out just right.

The trouble arises from the vanishing, in our reference matrix, of the two-rowed
determinant

$\big{|} \begin{array}{cc} 11 & 12 \\ 11^3 & 12^3 \end{array} \big{|} .$
Note that

$\big{|} \begin{array}{cc} 11 & 12 \\ 11^3 & 12^3 \end{array} \big{|} = 11\cdot 12\cdot \big{|} \begin{array}{cc} 1 & 1 \\ 11^2 & 12^2 \end{array} \big{|} = 17 \big{|} \begin{array}{cc} 1 & 1 \\ 11^2 & 12^2 \end{array} \big{|} =0,$
since $11^2=12^2$.

From the fact that $\big{|} \begin{array}{cc} 11 & 12 \\ 11^3 & 12^3 \end{array} \big{|}$ is the only vanishing determinant of any order in the matrix employed, all other assertions made in (1′) and (2′) are readily justified. This will be made clear in the following sections.

It will be advantageous, as shown more completely in subsequent sections, to employ reference matrices which contain the smallest possible number of vanishing determinants of any orders.