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The information in a code on $n$ neurons can also be encoded in an ideal generated by pseudo-monomials in $\mathbb{F}_2[x_1,...x_n]$.

A [**pseudo-monomial**]{} $f \in \mathbb{F}_2[x_1,...,x_n]$ is a polynomial with the form $$f = \prod_{i \in \sigma} x_i \prod_{j \in \tau} (1+x_j),$$ where $\sigma, \tau \in [n]$ and $\sigma \cap \tau = \emptyset$.

For any vector $v \in \mathbb{F}_2^n$, we define $$p_v = \prod_{i \in \text{supp(}v\text{)}} x_i \prod_{j \notin \text{supp(}v\text{)}}(1+x_j).$$ Such a pseudo-monomial is called the characteristic function for $v$ because $p_v(v) = 1$, but $p_v(x) = 0$ for all $x \in \mathbb{F}_2^n$ such that $x \neq v$. The neural ideal of a code is defined in terms of characteristic functions.

The [**neural ideal**]{} $J_C$ of a code $C$ is the ideal generated by all $p_v \in \mathbb{F}_2[x_1, ...,x_n]$ such that $v \notin C$. That is, $$J_C := \langle \{p_v \ | \ v \notin C \} \rangle.$$

Notice that this implies that if $f \in J_C$, then $f(c) = 0$ for any $c \in C$. In [@Neural_Rings], Curto et al. define the ideal of a code as follows:

Let $C \subseteq \mathbb{F}_2^n$ be a neural code. Then the [**ideal of 