By Barnette D.W.

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3. DIGITAL ARCHITECTURES V1 V2 node 1 node 2 25 X1 X2 node M Y1 .. ZJ . dec YJ B .. VM Z1 Vˆ1 Vˆ2 .. 2. This problem is often referred to as the multiple access channel, here with generally limited cooperation between the nodes and some forms of feedback. For the standard capacity problem, this question has been resolved by Ahlswede (1971) and Liao (1972) in the shape of the following theorem. 9 The capacity region of the multiple access channel with independent messages and without feedback and encoder cooperation is given by the convex closure of the union over all product distributions M p(x1 , x2 , .

PM satisfying M Pm ≤ Ptot (M). 10) m=1 The receiver observes these codewords across a Gaussian vector channel. d. (both over n and over j ) circularly symmetric complex Gaussian random variables of mean zero and variance σZ2 . 12) where Y[n] = (Y1 [n], Y2 [n], . . , YJ [n])T is a complex-valued column vector of length J , B (M) is a complex-valued matrix of dimensions J × M, with entries {B (M) }i,j = bi,j , X[n] = (X1 [n], X2 [n], . . , XM [n])T is a complex-valued column vector of length M, and Z[n] = (W1 [n], W2 [n], .

It can also be seen as a cut-set argument: We cut the network into two parts, one comprised of the nodes 1, 2, . . , M, the other comprised of the base station. More generally, one can consider arbitrary networks, and partition the nodes into two disjoint sets, S and S c . 12 (min-max cut-set bound) If it is feasible to communicate the sources (V1 , V2 , . . , VM ) across an M-user multiple access channel, then we must have H (VS ) ≤ CS for all S ⊆ {1, 2, . . 66) where the maximum is over all joint distributions p(xS,S c ) that satisfy the cost constraints P1 , .

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A 2-manifold of genus 8 without the W v-property by Barnette D.W.

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