OF LINEAR RECURRENCES AND LINEAR CODES OVER GALOIS RINGS

**A.S.KUZMIN, A.A.NECHAEV**

Let
*R*=*GR*(*q*^{2},4) be a Galois ring of the characteristic 4 with residue
field
*R*/2*R* =*GF*(*q*), *q*=2^{k}. A monic reversible polynomial *F*(*x*) of the
degree *m* over *R* is called distinquished if its period *T* over *R*equals to
;
and it is called a polynomial of maximal period if
.
Let *L*_{R}(*F*) be the family of all linear recurrences with the
characteristic polynomial *F*(*x*) and
be the set of initial segments
of all recurrences
.
Then
is a
linear code over *R*. The complete weight enumerator (c.w.e.) of such a code
is calculated. It gives the full description of possible types of
distributions of the ring *R* elements on cycles of the family *L*_{R}(*F*),
and quantity of the cycles of each given type. For example, if *T*=*q*^{m}-1 the
frequencies of elements
on cycles are described by the numbers
,
where
;
and quantities are described by the similar expressions.
These results in particulary allows to calculate the c.w.e. of generalized
Kerdock code over an arbitrary Galois field of characteristic 2. They are
based on the theory of quadrics over *GF*(2^{k}) and essentially precise the
estimation of *N*_{u}(*c*) by Kumar,Helleset,Calderbank (1995).

Centre of New Information Technologies

of Moscow Lomonosov State University

(e-mail: nechaev@cnit.chem.msu.su)

IMACS ACA'98 Electronic Proceedings