An important goal in communication theory is to construct good minimum squared Euclidean distance (MSED) codes for transmission over additive white Gaussian noise (AWGN) channels. In this paper, a new construction method for the M-ary phase-shift-keyed (M-PSK) codes over the ring structure ZM, the ring of integers modulo M, with a good minimum Euclidean distance, is proposed. The proposed codes are linear when multiple coset leaders are considered. The characteristics and performance levels of the newly constructed codes are analyzed for code length up to n 8. It is found that the proposed codes compare favorably with Piret's codes and Graeffe's method codes on Gaussian channels in terms of decoding complexity, coding gain, and error performance.

We discuss Nash equilibria in combinatorial auctions with item bidding. Specifically, we give a characterization for the existence of a Nash equilibrium in a combinatorial auction with item bidding when valuations by n bidders satisfy symmetric and subadditive properties. By this characterization, we can obtain an algorithm for deciding whether a Nash equilibrium exists in such a combinatorial auction.

In multilevel block modulation codes for QPSK and 8-PSK modulation, a construction of binary component codes is given. These codes have a good minimum Euclidean distance by using different forms of the dependency properties of the binary component codes. Interdependency among component codes is formed by using the binary component subcodes which are derived by the coset decomposition of the binary component codes. The algebraic structures of the codes are investigated to find out how interdependency among component codes gives a good minimum Euclidean distance. First, it is shown that cyclic codes over ZM for M-PSK (M=4,8), where the coding scheme is given by Piret, can be constructed by forming specific interdependency among binary component codes for proposed multilevel coding method. Furthermore, it is shown that better minimum Euclidean distance than above can be obtained by modifying the composition of interdependency among binary component codes. These proposed multilevel codes have algebraic structure of additive group and cyclic property over GF(M). Finally, error performances are compared with those of some code's reference modulation scheme for transmitting the same number of information bits.