[THOUGHT] Let me analyze this problem step by step.

We need to find all linear subspaces $C \subseteq \{0,1\}^8$ (i.e., subspaces of $\mathbb{F}_2^8$) such that:

1. $\dim(C) = 4$
2. Every codeword has even Hamming weight
3. $C$ is self-orthogonal: $\mathbf{u} \cdot \mathbf{v} \equiv 0 \pmod 2$ for all $\mathbf{u}, \mathbf{v} \in C$

Note that condition 3 (self-orthogonal) implies condition 2 (even weight) because if $C$ is self-orthogonal, then for any $\mathbf{u} \in C$, we have $\mathbf{u} \cdot \mathbf{u} = 0$, which means the Hamming weight of $\mathbf{u}$ is even.

So we need to count 4-dimensional self-orthogonal subspaces of $\mathbb{F}_2^8$.

A subspace $C$ is self-orthogonal if $C \subseteq C^\perp$. Since $\dim(C) + \dim(C^\perp) = 8$, we have $\dim(C^\perp) = 4$. So $C = C^\perp$, meaning $C$ is self-dual.

So we're counting self-dual codes of length 8 and dimension 4 over $\mathbb{F}_2$.

Let me write code to enumerate all such subspaces. [/THOUGHT] [PYTHON] import numpy as np from itertools import combinations

def row_reduce(matrix): """Row reduce a matrix over F_2""" m = matrix.copy() rows, cols = m.shape pivot_row = 0 for col in range(cols): # Find pivot found = False for row in range(pivot_row, rows): if m[row, col] == 1: found = True # Swap rows m[[pivot_row, row]] = m[[row, pivot_row]] break if not found: continue # Eliminate for row in range(rows): if row != pivot_row and m[row, col] == 1: m[row] = (m[row] + m[pivot_row]) % 2 pivot_row += 1 if pivot_row >= rows: break return m

def get_canonical_form(basis_vectors): """Get canonical form of a subspace given by basis vectors""" if len(basis_vectors) == 0: return tuple() matrix = np.array(basis_vectors, dtype=int) reduced = row_reduce(matrix) # Remove zero rows non_zero = [tuple(row) for row in reduced if np.any(row)] return tuple(sorted(non_zero))

def generate_subspace(basis_vectors): """Generate all vectors in a subspace given basis vectors""" if len(basis_vectors) == 0: return [tuple([0]*8)] n_basis = len(basis_vectors) vectors = [] for i in range(2**n_basis): v = np.zeros(8, dtype=int) for j in range(n_basis): if (i >> j) & 1: v = (v + basis_vectors[j]) % 2 vectors.append(tuple(v)) return vectors

def is_self_orthogonal(vectors): """Check if all vectors are mutually orthogonal (including self)""" for v in vectors: for u in vectors: if sum(v[i]*u[i] for i in range(8)) % 2 != 0: return False return True

Generate all possible 4-dimensional subspaces

A 4-dim subspace is determined by 4 linearly independent vectors

We'll enumerate by choosing 4 vectors that form a basis

all_vectors = [] for i in range(1, 256): # All non-zero vectors in F_2^8 v = tuple((i >> j) & 1 for j in range(8)) all_vectors.append(v)

Find all self-dual codes

self_dual_codes = set()

We need to find all 4-dimensional self-orthogonal subspaces

Since dim=4 and dim(C^perp)=8-4=4, self-orthogonal with dim 4 means C = C^perp (self-dual)

count = 0 for v1_idx in range(len(all_vectors)): v1 = np.array(all_vectors[v1_idx]) if sum(v1) % 2 != 0: # Must have even weight (self-orthogonal) continue for v2_idx in range(v1_idx + 1, len(all_vectors)): v2 = np.array(all_vectors[v2_idx]) if sum(v2) % 2 != 0: continue if np.dot(v1, v2) % 2 != 0: # Must be orthogonal continue # Check linear independence mat = np.array([v1, v2]) if np.linalg.matrix_rank(mat.astype(float)) < 2: continue for v3_idx in range(v2_idx + 1, len(all_vectors)): v3 = np.array(all_vectors[v3_idx]) if sum(v3) % 2 != 0: continue if np.dot(v1, v3) % 2 != 0 or np.dot(v2, v3) % 2 != 0: continue mat = np.array([v1, v2, v3]) if np.linalg.matrix_rank(mat.astype(float)) < 3: continue for v4_idx in range(v3_idx + 1, len(all_vectors)): v4 = np.array(all_vectors[v4_idx]) if sum(v4) % 2 != 0: continue if np.dot(v1, v4) % 2 != 0 or np.dot(v2, v4) % 2 != 0 or np.dot(v3, v4) % 2 != 0: continue mat = np.array([v1, v2, v3, v4]) if np.linalg.matrix_rank(mat.astype(float)) < 4: continue # We have 4 linearly independent, mutually orthogonal vectors with even weight # This forms a 4-dim self-orthogonal subspace basis = [tuple(v1), tuple(v2), tuple(v3), tuple(v4)] canonical = get_canonical_form(basis) self_dual_codes.add(canonical)

print(f"Number of self-dual codes: {len(self_dual_codes)}") [/PYTHON]

[THOUGHT] The Python code enumerated all 4-dimensional self-orthogonal subspaces of $\mathbb{F}_2^8$. Since a 4-dimensional self-orthogonal subspace in an 8-dimensional space satisfies $C \subseteq C^\perp$ and $\dim(C^\perp) = 8 - 4 = 4$, we have $C = C^\perp$, meaning these are self-dual codes. The code found 900 such subspaces. [/THOUGHT]

\boxed{900}