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.

For \( n \in \mathbb{N} \), we interpret the vertex set \( W_n \) of the \( n \)-cube as a vector space over the field \( \mathbb{F}_2 \) and prove that a regular \( n \)-simplex can be inscribed into the \( n \)-cube such that its vertices constitute a subgroup of \( W_n \) if and only if \( n+1 \) is a power of 2. Furthermore, a connection to the theory of Hamming Codes will be established.

An \( (n,k) \) binary self-orthogonal code is an \( (n,k) \) binary linear code \( C \) that is contained in its orthogonal complement \( C^\bot \). A self-orthogonal code \( C \) is self-dual if \( C = C^\bot \). Two codes, \( C_1 \) and \( C_2 \), are \({equivalent}\) if and only if there exists a coordinate permutation of \( C_1 \) that takes \( C_1 \) into \( C_2 \). The automorphism group of a code \( C \) is the set of all coordinate permutations of \( C \) that takes \( C \) into itself.

This paper is a continuation of the work presented in [2], in which we described an algorithm for enumerating inequivalent binary self-dual codes. We used our algorithm to enumerate the self-dual codes of length up to and including 32. Our algorithm also found the size of the automorphism group of each code.

We have since made several improvements to our algorithm. It now generally runs faster. It also now finds generators for the automorphism group of each code. We have used our improved algorithm to enumerate the self-dual codes of length 34. We have also found the automorphism groups for each of our self-dual codes of length less than or equal to 34. The list of length 34 codes are new, as are the lists of automorphism groups for the length 32 and length 34 codes. We have found there are 19914 inequivalent length 34 codes with distance 4 and 938 length 34 codes with distance 6.

A graph is claw-free if it has no induced \( K_{1,3} \) subgraph. A graph is essential 4-edge-connected if removing at most three edges, the resulting graph has at most one component having edges. In this note, we show that every essential 4-edge-connected claw-free graph has a spanning Eulerian subgraph with maximum degree at most 4.

A labeling \( f \) of a graph \( G \) is called semi-H-cordial if for each vertex \( v \), \( |f(v)| \leq 1 \), \( |e_f(1) – e_f(-1)| \leq 1 \) and \( |v_f(1) – v_f(-1)| \leq 1 \). In this paper we study the forcing semi-H-cordial numbers of paths, cycles, stars, trees, Dutch-windmill graphs, wheels, grids and cylinders.

A three-fold Kirkman packing design \( \text{KPD}_3(\{4,s^*\},v) \) is a three-fold resolvable packing with maximum possible number of parallel classes, each containing one block of size 3 and all other blocks of size 4. This article investigates the spectra of three-fold Kirkman packing design \( \text{KPD}_3(\{4,s^*\},v) \) for \( s = 5 \) and \( 6 \), and we show that it contains all positive integers \( v \equiv s – 4 \pmod{4} \) with \( v \geq 17 \) if \( s = 5 \), and \( v \geq 26 \) if \( s = 6 \).

Let \( G \) be a \( (p,q) \)-graph in which the edges are labeled \( 1, 2, 3, \ldots, q \). The vertex sum for a vertex \( v \) is the sum of the labels of the incident edges at \( v \). If \( G \) can be labeled so that the vertex sums are distinct, mod \( p \), then \( G \) is said to be edge-graceful. If the edges of \( G \) can be labeled \( 1, 2, 3, \ldots, q \) so that the vertex sums are constant, mod \( p \), then \( G \) is said to be edge-magic. It is conjectured by Lee [9] that any connected simple \( (p,q) \)-graph with \( q(q+1) \equiv p(p-1)/2 \pmod{p} \) vertices is edge-graceful. We show that the conjecture is true for maximal outerplanar graphs. We also completely determine the edge-magic maximal outerplanar graphs.