On proper secrets, (t, k)-bases and linear codes

This paper contains three parts where each part triggered and motivated the subsequent one. In the first part (Proper Secrets) we study the Shamir's "k-out-of-n" threshold secret sharing scheme. In that scheme, the dealer generates a random polynomial of degree k-1 whose free coefficient is the secret and the private shares are point values of that polynomial. We show that the secret may, equivalently, be chosen as any other point value of the polynomial (including the point at infinity), but, on the other hand, setting the secret to be any other linear combination of the polynomial coefficients may result in an imperfect scheme. In the second part ((t, k)-bases) we define, for every pair of integers t and k such that 1 ≤ t ≤ k-1, the concepts of (t, k)-spanning sets, (t, k)-independent sets and (t, k)-bases as generalizations of the usual concepts of spanning sets, independent sets and bases in a finite-dimensional vector space. We study the relations between those notions and derive upper and lower bounds for the size of such sets. In the third part (Linear Codes) we show the relations between those notions and linear codes. Our main notion of a (t, k)-base bridges between two well-known structures: (1, k)-bases are just projective geometries, while (k-1, k)-bases correspond to maximal MDS-codes. We show how the properties of (t, k)-independence and (t, k)-spanning relate to the notions of minimum distance and covering radius of linear codes and how our results regarding the size of such sets relate to known bounds in coding theory. We conclude by comparing between the notions that we introduce here and some well known objects from projective geometry.