A Class of Arithmetic Codes With Correction of Several Errors

Let $p$ be a prime number and 2 a primitive root of it. Then the vector $a^{(0)}$ of dimension $p- 1$ generates by means of a cyclic shift a group of $p$ vectors with the operation of addition mod $2^{p-1}-1$, where $a^{(0)}$ is the binary notation of the number $(2^{p-1}-1)/p$. This group is a code correcting several errors.