Studies on the distribution of the shortest linear recurring sequences

The distribution of the shortest linear recurrence (SLR) sequences in the Z/(p) field and over the Z/(pe) ring is studied. It is found that the length of the shortest linear recurrent (SLRL) is always equal to n/2, if n is even and n/2 + 1 if n is odd in the Z/(p) field, respectively. On the other hand, over the Z/(pe) ring, the number of sequences with length n can also be calculated. The recurring distribution regulation of the shortest linear recurring sequences is also found. To solve the problem of calculating the SLRL, a new simple representation of the Berlekamp–Massey algorithm is developed as well.