Several classes of minimal binary linear codes violating the Ashikhmin-Barg bound

Abstract

Minimal binary linear codes are a special class of binary codes with important applications in secret sharing and secure two-party computation. These codes are characterized by the property that none of the nonzero codewords is covered by any other codeword. Denoting by $w_{{\min \limits }}$ and $w_{{\max \limits }}$ the minimum and maximum weights of the codewords, respectively, such codes are relatively easy to design when the ratio $w_{{\min \limits }}/w_{{\max \limits }}$ is larger than 1/2 (known as the Ashikhmin-Barg bound). On the other hand, a few known classes of minimal codes violate this bound, hence having the property $w_{{\min \limits }}/w_{{\max \limits }} \leq 1/2$. In this article, we provide several explicit classes of minimal binary linear codes that violate the Ashikhmin-Barg bound while achieving a great variety of the ratio $w_{{\min \limits }}/w_{{\max \limits }}$. Our first generic method employs suitable characteristic functions with relatively low weights within the range [n + 1,2^n− 2]. The second approach specifies characteristic functions with weights in [2^n− 2 + 1,2^n− 2 + 2^n− 3 − 1], whose supports contain a skewed (removing one element) affine subspace of dimension n − 2. Finally, we also characterize an infinite family of minimal codes based on the class of so-called root Boolean functions of weight 2^n− 1 − (n − 1), useful in specific hardware testing applications. Consequently, many infinite classes of minimal codes crossing the Ashikhmin-Barg bound are derived from an ample range of characteristic functions. In certain cases, we completely specify the weight distributions of the resulting codes.