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All inductively pierced codes are nondegenerate convex codes and nondEGenerate hyperplane codes, and it is proved that a $k-inductively pierced code on $n$ neurons has a convex realization with balls in $\mathbb R^{k+1}$ and with half spaces in $n}$.

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Giusti and Itskov prove that convex neural codes have no "local obstructions," which are defined via the topology of a code's simplicial complex, and reveal a stronger type of local obstruction that prevents a code from being convex, and prove that the corresponding decision problem is NP-hard.

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### Inductively pierced codes and neural toric ideals

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All inductively pierced codes are nondegenerate convex codes and nondEGenerate hyperplane codes, and it is proved that a $k-inductively pierced code on $n$ neurons has a convex realization with balls in $\mathbb R^{k+1}$ and with half spaces in $n}$.

### Gröbner bases of neural ideals

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It is proved that if the canonical form of a neural ideal is a Gr\"obner basis, then it is the universal Gr\"OBner basis (that is, the union of all reduced Gr \"obner bases).

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### Morphisms of Neural Codes

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