As for the definitions and notation, we refer to our paper with Prof.Freitag: "A theta relation in genus 4", Nagoya Math. J.(2001). We take the binary self-dual doubly-even codes length 32 from Professor Sloane's home page (http://www.research.att.com/~njas/codes/CP.32.16.txt). We call them C1, C2,..., C85.

For a code C, we denote by W(C) the weight enumerator of the code C in genus 4.

The dimension of the vector space of the weight enumerators of length 32 in genus 4 is 19. We take, as the basis of this vector space, W(C1),W(C2),W(C3),W(C4),W(C5),W(C6),W(C7), W(C10),W(C11),W(C16),W(C18),W(C23), W(C24),W(C25),W(C27),W(C29),W(C44),W(C67),W(C82).

admissible monomials
There exist 1083 admissible monomials of Weight 32 in genus 4 upto AGL. They are given by the 1083 x 16 matrix. We take 0000,0001,0010,...,1111 as the ordering of F_2^4 if we need. For example, the first row [0,0,4,2,...,0] means x0000^0*x0001^0*x0010^4*x0011^2*...*x1111^0.

length of admissible monomials
The 1083 numbres are the lengths of the orbits AGL*v, where v is one of the admissible monomials. The ordering of the follwing corresponds to that of the admissible monomials. For example, the first 40320 means the length of the orbits AGL*v is 40320, where v=[ 0, 0, 4, 2, 8, 0, 4, 2, 4, 2, 4, 0, 0, 2, 0, 0 ].

coefficients of the basis(19 codes)
We give the coefficients of the weight enumerators of the basis W(C1),W(C2),W(C3),W(C4),W(C5),W(C6),W(C7), W(C10),W(C11),W(C16),W(C18),W(C23), W(C24),W(C25),W(C27),W(C29),W(C44),W(C67),W(C82). This is given by the 19 x 1083 matrix. For example, the first row [ 54486432000,...] means W(C1)=54486432000*x0000^0*x0001^0*x0010^4*x0011^2*...*x1111^0+...

relations
Each entry a(i,j) of the 85 x 19 matrix all=(a(i,j)) is determined as follows. For each i, 0<=i<=85, we write W(Ci)=
a(i,1)*W(C1)+
a(i,2)*W(C2)+
a(i,3)*W(C3)+
a(i,4)*W(C4)+
a(i,5)*W(C5)+
a(i,6)*W(C6)+
a(i,7)*W(C7)+
a(i,8)*W(C10)+
a(i,9)*W(C11)+
a(i,10)*W(C16)+
a(i,11)*W(C18)+
a(i,12)*W(C23)+
a(i,13)*W(C24)+
a(i,14)*W(C25)+
a(i,15)*W(C27)+
a(i,16)*W(C29)+
a(i,17)*W(C44)+
a(i,18)*W(C67)+
a(i,19)*W(C82).

coefficients for all 85 codes
We give the coefficients of W(C1),W(C2),...,W(C85).