The "neural code" is the way the brain characterizes, stores, and processes information. Unraveling the neural code is a key goal of mathematical neuroscience. Topology, coding theory, and, recently, commutative algebra are some the mathematical areas that are involved in analyzing these codes. Neural rings and ideals are algebraic objects that create a bridge between mathematical neuroscience and commutative algebra. A neural ideal is an ideal in a polynomial ring that encodes the combinatorial firing data of a neural code. Using some algebraic techniques one hopes to understand more about the structure of a neural code via neural rings and ideals. In this paper, we introduce an operation, called "polarization," that allows us to relate neural ideals with squarefree monomial ideals, which are very well studied and known for their nice behavior in commutative algebra.2010 Mathematics Subject Classification. 13F20, 13P25, 13L99, 92B20.

In this paper, we introduce the k × n (with k ≤ n) truncated, supplemented Pascal matrix which has the property that any k columns form a linearly independent set. This property is also present in Reed-Solomon codes; however, Reed-Solomon codes are completely dense, whereas the truncated, supplemented Pascal matrix has multiple zeros. If the maximal-distance separable code conjecture is correct, then our matrix has the maximal number of columns (with the aformentioned property) that the conjecture allows. This matrix has applications in coding, network coding, and matroid theory.

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We formulate an Algebraic-Coding Equivalence to the Maximum Distance Separable Conjecture. Specifically, we present novel proofs of the following equivalent statements. Let (q, k) be a fixed pair of integers satisfying q is a prime power and 2 ≤ k ≤ q. We denote by P q the vector space of functions from a finite field F q to itself, which can be represented as the space P q := F q [x]/(x q − x) of polynomial functions. We denote by O n ⊂ P q the set of polynomials that are either the zero polynomial, or have at most n distinct roots in F q . Given two subspaces Y, Z of P q , we denote by Y, Z their span. We prove that the following are equivalent.A Suppose that either:(a) q is odd (b) q is even and k ∈ {3, q − 1}.Then there do not exist distinct subspaces Y and Z of P q such that:B Suppose q is odd, or, if q is even, k ∈ {3, q − 1}. There is no integer s with q ≥ s > k such that the Reed-Solomon code R over F q of dimension s can have s − k + 2 columns B = {b 1 , . . . , b s−k+2 } added to it, such that:(a) Any s × s submatrix of R ∪ B containing the first s − k columns of B is independent. (b) B ∪ {[0, 0, . . . , 0, 1]} is independent.C The MDS conjecture is true for the given (q, k).