Equality conjecture for the codes ${\mathbb C}(m,j)$ and ${\mathbb C}_{2^j}^{{\cal H}_m}$

Let ${\mathbb C}(m,j)$ be the code defined earlier in the source, and let ${\mathbb C}_{2^j}^{{\cal H}_m}$ be the code associated with the Hadamard matrix ${\cal H}_m$ and modulus $2^j$. Equality conjecture for the two code constructions.

$${\mathbb C}(m,j)={\mathbb C}_{2^j}^{{\cal H}_m}.$$

The preceding lemma establishes only the inclusion ${\mathbb C}(m,j)\subseteq {\mathbb C}_{2^j}^{{\cal H}_m}$, so the conjecture asserts that this inclusion is equality; the source gives no resolution.