Key distributions based on cover-free families have been recently proposed for secure rekeying in group communication systems after multiple simultaneous user ejections. Existing literature has not quantified how difficult this rekeying operation might be. This study provides upper bounds on the number messages necessary to rekey a key distribution based on symmetric combinatorial designs after one or two simultaneous user ejections. Connections are made to results from finite geometry to show that these bounds are tight for certain key distributions. It is shown that in general determining the minimal number of messages necessary to rekey a group communication system based on a cover-free family is NP-hard.

Complex orthogonal designs (CODs) of rate 1/2 have been considered recently for use in analog transmissions and as an alternative to maximum rate CODs due to the savings in decoding delay as the number of antennas increases. While algorithms have been developed to show that an upper bound on the minimum decoding delay for rate 1/2 CODs withn=2m-1 orn=2mcolumns is ν(n) = 2m-1 or ν(n) = 2m, depending on the parity ofnmodulo 8, it remains open to determine the exact minimum delay. This paper shows that this bound ν(n) is also a lower bound on minimum decoding delay for a major class of rate 1/2 CODs, named balanced complex orthogonal designs (BCODs), and that this is the exact minimum decoding delay for most BCODs. These rate 1/2 codes are conjugation-separated and thus permit a linearized description of the transceiver signal. BCODs also display other combinatorial properties that are expected to be useful in implementation, such as having no linear processing. An elegant construction is provided for a class of rate 1/2 CODs that have no zero entries, effectively no irrational coefficients, no linear processing, and have each variable appearing exactly twice per column. The resulting codes meet the aforementioned bound on decoding delay in most cases. This class of CODs will be useful in practice due to their low peak-to-average power ratio (PAPR) and other desirable properties.

Complex orthogonal designs (CODs) have been successfully implemented in wireless systems as complex orthogonal space-time block codes (COSTBCs). Certain properties of the underlying CODs affect the performance of the codes. In addition to the main properties of a COD's rate and decoding delay, a third consideration is whether the COD can achieve transceiver signal linearization, a property that facilitates practical implementation by, for example, significantly simplifying the receiver structure for iterative decoding. It has been shown that a COD can achieve this transceiver signal linearization if the nonzero entries in any given row of the matrix are either all conjugated or all nonconjugated. This paper determines the conditions under which maximum rate CODs can achieve this desirable property. For an odd number of transmit antennas, it is shown that maximum rate CODs that achieve the lower bound on decoding delay can also achieve transceiver signal linearization. In contrast, for an even number of transmit antennas, maximum rate CODs that achieve the lower bound on delay cannot achieve this linearization. In this latter case, linearization is possible only if the COD achieves at least twice the lower bound on delay. This work highlights the tradeoffs among these three important properties.

Complex orthogonal space-time block codes (COSTBCs) based on generalized complex orthogonal designs (CODs) have been successfully implemented in wireless systems with multiple transmit antennas and single or multiple receive antennas. It has been shown that for a maximum rate COD with 2m-1 or 2m columns, a lower bound on decoding delay is (m-12m) and this delay is achievable when the number of columns is congruent to 0, 1 , or 3 modulo 4. In this paper, the final case is addressed, and it is shown that when the number of columns is congruent to 2 modulo 4, the lower bound on decoding delay cannot be achieved. In this case, the shortest decoding delay a maximum rate COD can achieve is twice the lower bound. New techniques for analyzing CODs are introduced with connections to binary vector spaces.

The growing demand for efficient wireless transmissions over fading channels motivated the development of space-time block codes. Space-time block codes built from generalized complex orthogonal designs are particularly attractive because the orthogonality permits a simple decoupled maximum-likelihood decoding algorithm while achieving full transmit diversity. The two main research problems for these complex orthogonal space-time block codes (COSTBCs) have been to determine for any number of antennas the maximum rate and the minimum decoding delay for a maximum rate code. The maximum rate for COSTBCs was determined by Liang in 2003. This paper addresses the second fundamental problem by providing a tight lower bound on the decoding delay for maximum rate codes. It is shown that for a maximum rate COSTBC for 2m - 1 or 2m antennas, a tight lower bound on decoding delay is r = (m-12m) . This lower bound on decoding delay is achievable when the number of antennas is congruent to 0, 1, or 3 modulo 4. This paper also derives a tight lower bound on the number of variables required to construct a maximum rate COSTBC for any given number of antennas. Furthermore, it is shown that if a maximum rate COSTBC has a decoding delay of r where rm-12m_).