In the Shamir \( (k,n) \) threshold scheme, if one or more of the \( n \) shares are fake, then the secret may not be reconstructed correctly by some sets of \( k \) shares. Supposing that at most \( t \) of the \( n \) shares are fake, Rees et al. (1999) described two algorithms to determine consistent sets of shares so that the secret can be reconstructed correctly from \( k \) shares in any of these consistent sets. In their algorithms, no honest participant can be absent and at least \( n – t \) shares should be pooled during the secret reconstruction phase. In this paper, we propose a modified algorithm for this problem so that the number of participants taking part in the secret reconstruction can be reduced to \( k + 2t \) and the shares need to be pooled can be reduced to, in the best case, \( k + t \), and less than or equal to \( k + 2t \) in the others. Its performance is evaluated. A construction for \( t \)-coverings, which play key roles in these algorithms, is also provided.

A linear \( k \)-forest is a graph whose components are paths with lengths at most \( k \). The minimum number of linear \( k \)-forests needed to decompose a graph \( G \) is the linear \( k \)-arboricity of \( G \) and is denoted by \( la_k(G) \). In this paper, we study the linear \( 3 \)-arboricity of balanced complete multipartite graphs and we obtain some substantial results.

Let \( m_2(N, q) \) denote the size of the largest caps in \( PG(N, q) \) and let \( m_2′(N, q) \) denote the size of the second-largest complete caps in \( PG(N, q) \). Presently, it is known that \( m_2(4, 5) \leq 111 \) and that \( m_2(4, 7) \leq 316 \). Via computer searches for caps in \( PG(4, 5) \) using the result of Abatangelo, Larato, and Korchmáros that \( m_2′(3, 5) = 20 \), we improve the first upper bound to \( m_2(4, 5) \leq 88 \). Computer searches in \( PG(3, 7) \) show that \( m_2′(3, 7) = 32 \), and this latter result then improves the upper bound on \( m_2(4, 7) \) to \( m_2(4, 7) \leq 238 \). We also present the known upper bounds on \( m_2(N, 5) \) and \( m_2(N, 7) \) for \( N > 4 \).

A sum of disjoint products (SDP) representation of a Boolean function is useful because it provides readily available information about the function; however, a typical SDP contains many more terms than an equivalent ordinary sum of products. We conjecture the existence of certain particular SDP forms of \( x_1 + \cdots + x_t \), which could be used as patterns in creating relatively economical SDP forms of other Boolean functions.