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).