On the Error Detection Capability of One Check Digit

In this paper, we study a check digit system which is based on the use of elementary abelian p-groups of order p
^k
. This paper is inspired by a recently introduced check digit system for hexadecimal numbers. By interpreting its check equation in terminology of matrix algebra, we generalize the idea to build systems over a group of order p
^k
, while keeping the ability to detect all the: 1) single errors; 2) adjacent transpositions; 3) twin errors; 4) jump transpositions; and 5) jump twin errors. Besides, we consider two categories of jump errors: 1) t-jump transpositions and 2) t-jump twin errors, which include and further extend the double error types of 2)-5). In particular, we explore R
_c
, the maximum detection radius of the system on detecting these two kinds of generalized jump errors, and show that it is 2
^k
-2 for p=2 and (p
^k
-1)/2-1 for an odd prime p. Also, we show how to build such a system that detects all the single errors and these two kinds of double jump-errors within R
_c
.