Summary: Broadcasting is the task of transmitting a message originated at one processor of a communication network to all other processors in the network. A minimal \(k\)-fault-tolerant broadcast network is a communication network on \(n\) vertices in which any processor can broadcast in spite of up to \(k\) line failures in optimal time \(T_n(k)\). In this paper, we study \(B_k(n)\), the minimum number of communication lines of any minimal \(k\)-fault-tolerant broadcast network on \(n\) processors. We give the value of \(B_k(n)\) for several values of \(n\) and \(k\) and, in case \(k<\lfloor\log n\rfloor\), give almost-minimum \(k\)-fault-tolerant broadcast networks.