A deterministic coding scheme is presented for additive white Gaussian broadcast channels with two receivers and feedback from the receivers to the transmitter. The region of data rates at which reliable communication is possible is larger than that of the corresponding channel without feedback. This is the first example showing that the capacity region of degraded broadcast channels can be enlarged with feedback.

A recent paper [1] discussed the2^{-p}bound (wherep = n- k) for the probability of undetected errorP(\epsilon)for an(n,k)block code used for error detection on a binary symmetric channel. This investigation is continued and extended and dual codes are studied. The dual and extension of a perfect code obey the2^{-p}bound, but this is not necessarily true for arbitrary codes that obey the bound. Double-error-correcting BCH codes are shown to obey the bound. Finally the problem of constructing uniformly good codes is examined.

It is known that the expected codeword lengthL_{UD}of the best uniquely decodable (UD) code satisfiesH(X)\leqL_{UD}

Wyner's results for discrete memoryless wire-tap channels are extended to the Gaussian wire-tap channel. It is shown that the secrecy capacity Cs is the difference between the capacities of the main and wire.tap channels. It is further shown that Rd= Cs is the upper boundary of the achievable rate-equivocation region.

The special class of wiretap channels for which a certain quantity\Gamma(R)is constant is examined, and some useful characterizations of its members are given. It is shown that for "symmetric" wiretap channels,\Gamma(R)is equal to the difference between the capacities of the main channel and the channel from the sender to the wiretapper.

In the past, it has generally been assumed that the probability of undetected error for an(n,k)block code, used solely for error detection on a binary symmetric channel, is upperbounded by2^{-(n-k)}. In this correspondence, it is shown that Hamming codes do indeed obey this bound, but that the bound is violated by some more general codes. Examples of linear, cyclic, and Bose-Chaudhuri-Hocquenghem (BCH) codes which do not obey the bound are given.