A construction of maximal linearly independent array

Abstract

Let M be a subset of r-dimensional vector space V r (\( \mathbb{F} \) 2) over a finite field \( \mathbb{F} \) 2, consisting of n nonzero vectors, such that every t vectors of M are linearly independent over \( \mathbb{F} \) 2. Then M is called (n, t)-linearly independent array of length n over V r (\( \mathbb{F} \) 2). The (n, t)-linearly independent array M that has the maximal number of elements is called the maximal (r, t)-linearly independent array, and the maximal number is denoted by M(r, t). It is an interesting combinatorial structure, which has many applications in cryptography and coding theory. It can be used to construct orthogonal arrays, strong partial balanced designs. It can also be used to design good linear codes. In this paper, we construct a class of maximal (r, t)-linearly independent arrays of length r + 2, and provide some enumerator theorems.