A complex orthogonal design (COD) with parameter [p, n, k] is a combinatorial design used in space-time block codes (STBCs). For STBCs, n is the number of antennas, k/p is the rate, and p is the decoding delay. A class of rate 1/2 CODs called balanced complex orthogonal designs (BCODs) has been proposed by Adams et al., who constructed BCODs with rate k/p = 1/2 and decoding delay p = 2mfor n = 2m. Furthermore, they proved that the constructions have optimal decoding delay when m is congruent to 1, 2, or 3 modulo 4. They conjectured that for the case m ≡ 0 (mod 4), 2mis also a lower bound on p. In this paper, we prove this conjecture.

Complex orthogonal designs (CODs) are used to construct space-time block codes in wireless transmission. COD Ozwith parameter [p, n, k] is a p × n matrix, where nonzero entries are filled by ±zior ±zi* , i = 1, 2, ..., k, such that equation. In practice, n is the number of antennas, k=p the code rate, and p the decoding delay. One fundamental problem is to construct COD to maximize k/p and minimize p when n is given. Recently, this problem is completely solved by Liang and Adams et al. It's proved that when n = 2m or 2m - 1, the maximal possible rate is (m + 1)/(2m) and the minimum delay (m-12m)(with the only exception n ≡2 (mod 4) where it is 2(m-12m)). However, when the number of antennas increase, the minimum delay grows fast and eats the otherwise fast decoding. For example, when n = 14 the minimal delay for a code with maximal rate is 6006! Therefore, it is very important to study whether it is possible, by lowering the rate slightly, to shorten the decoding delay considerably. In this paper, we demonstrate this possibility by constructing a series of CODs with parameter [p, n, k] = [(w - 1n)+(w + 1n), n, (wn)], where 0 ≤ w ≤ n. Besides that, all optimal CODs, which achieve the maximal rate and minimal delay, are contained in our explicit-form constructions. And this is the first explicit-form construction, while the previous are recursive or algorithmic.

Complex orthogonal designs (CODs) are used to construct space-time block codes. COD Ozwith parameter [p, n, k] is a p × n matrix, where nonzero entries are filled ±ziby ±zi*, i = 1,2...,k, or , such that OzHOz= (|z1|2+ |z2|2+ ...+ |zk|2)In×n. Define Oza first type COD if and only if Ozdoes not contain submatrix (±zj,0:0,±zj*) or (±zj*,0:0,±zj). It is already known that all CODs with maximal rate, i.e., maximal k/p, are of the first type. In this paper, we will determine all achievable parameters [p, n, k] of first type COD, as well as all their possible structures. The existence of parameters is proved by explicit-form constructions. New CODs with parameters [p, n, k] = [(n:w-1) + (n:w+1),n, (n:w)], for 0 ≤ w ≤ n, are constructed, which demonstrate the possibility of sacrificing code rate to reduce decoding delay. It is worth mentioning that all maximal rate, minimal delay CODs are contained in our constructions, and their uniqueness under equivalence operation is proved.

Given a positive even integer n, it is found that the weight distribution of any n-variable symmetric Boolean function with maximum algebraic immunity (AI) n/2 is determined by the binary expansion of n . Based on the foregoing, all n-variable symmetric Boolean functions with maximum AI are constructed. The amount is (2 wt(n)+1)2[log2n].

In this paper we investigate the problems of searching for the capacity factors and determining the capacity ranks of edges in a network coding-based network, which were first proposed in (K. Cai and P.Y. Fan, 2007). For the former problem, we prove that it is computationally hard by reducing the well known NP-complete SUB-SUM problem to the current problem. For the latter problem, we devise efficient algorithms in a special case of networks and conjecture that in general case the problem is also hard.