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Toric codes are evaluation codes obtained from an integral convex polytope P ⊂ R n and finite field Fq. They are, in a sense, a natural extension of Reed-Solomon codes, and have been studied recently in [6], [8], [9], and [12]. In this paper, we obtain upper and lower bounds on the minimum distance of a toric code constructed from a polygon P ⊂ R 2 by(More)
Hansen (Appl. Algebra Eng. Comm. Comput. 14 (2003) 175) uses cohomological methods to find a lower bound for the minimum distance of an evaluation code determined by a reduced complete intersection in P 2. In this paper, we generalize Hansen's results from P 2 to P m ; we also show that the hypotheses of Hansen (2003) may be weakened. The proof is succinct(More)
The minimal free resolution of the Jacobian ideals of the determinant polynomial were computed by Lascoux [12], and it is an active area of research to understand the Jacobian ideals of the permanent, see e.g., [13, 9]. As a step in this direction we compute several new cases and completely determine the linear strands of the minimal free resolutions of the(More)
In [19], Migliore–Miró-Roig–Nagel show that the Weak Lefschetz property can fail for an ideal I ⊆ K[x 1 ,. .. , x 4 ] generated by powers of linear forms. This is in contrast to the analogous situation in K[x 1 , x 2 , x 3 ], where WLP always holds [24]. We use the inverse system dictionary to connect I to an ideal of fat points, and show that failure of(More)
The method of shifted partial derivatives introduced in [9, 7] was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this method alone cannot prove that the padded permanent ℓ n−m perm m cannot be realized inside the GL n 2-orbit closure of the determinant detn when n > 2m 2 + 2m.(More)
Let U ⊆ H 0 (O P 1 ×P 1 (2, 1)) be a basepoint free four-dimensional vector space. The sections corresponding to U determine a regular map φ U : P 1 × P 1 −→ P 3. We study the associated bigraded ideal I U ⊆ k[s, t; u, v] from the standpoint of commutative algebra, proving that there are exactly six numerical types of possible bigraded minimal free(More)