Hyperplane Neural Codes and the Polar Complex

Hyperplane codes are a class of convex codes that arise as the output of a one layer feed-forward neural network. Here we establish several natural properties of stable hyperplane codes in terms of the {\it polar complex} of the code, a simplicial complex associated to any combinatorial code. We prove that the polar complex of a stable hyperplane code is shellable and show that most currently known properties of the hyperplane codes follow from the shellability of the appropriate polar complex.Comment: 23 pages, 5 figures. To appear in Proceedings of the Abel Symposiu

Maximum Distance Separable Codes and Arcs in Projective Spaces

Given any linear code $C$ over a finite field $GF(q)$ we show how $C$ can be described in a transparent and geometrical way by using the associated Bruen-Silverman code. Then, specializing to the case of MDS codes we use our new approach to offer improvements to the main results currently available concerning MDS extensions of linear MDS codes. We also sharply limit the possibilities for constructing long non-linear MDS codes.Comment: 18 Pages; co-author added; some results updated; references adde