WEIGHT CHARACTERISTICS
OF LINEAR RECURRENCES AND LINEAR CODES OVER GALOIS RINGS

A.S.KUZMIN, A.A.NECHAEV

Let R=GR(q²,4) be a Galois ring of the characteristic 4 with residue field R/2R =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 Requals to $\tau=q^m-1$; and it is called a polynomial of maximal period if $T= 2\tau$. Let L_R(F) be the family of all linear recurrences with the characteristic polynomial F(x) and $\cal{K}$ be the set of initial segments $u(\overline{0,T-1})$ of all recurrences $u \in L(F)$. Then $\cal{K}$ 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 $c \in R$ on cycles are described by the numbers $N_u(c)= q^{m-2}\pm wq^{\lambda -1}-\delta_{c,0}$, where $w \in \{0,1,q-1\}, ~\lambda=[m/2]$; 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)