Papers summery on: R.Zamir and S.Shamai and U.Erez, ”Nested linear/lattice codes for structured multiterminal binning”

In [1] nested codes are defined and some fundamental properties are introduced. We review the usage of nested codes in wyner’s noiseless binning scheme,noisy side information problems and dirty paper problems. Some related topic such as multiterminal coding of correlated sources and coordinated encoding over mutually interfering channels are also touched. I. NESTED CODES: PRELIMINARIES Nested code is a pair of linear or lattice codes (C1, C2) satisfying C2 ∈ C1. C1, C2 are called “fine code” and “coarse code” respectively. A. Nested Parity-Check Codes The check matrices of the parity-check codes of a nested code {(n, k1), (n, k2)}can be represent by H2 =   H1 · · · ∆H   The syndromes s1 = H1x and s2 = H2x satisfy s2 = [s t 1,∆s t]. When x is a codeword of C1, s2 = [0, · · · ,0,∆st]. We can partition C1 into 2k1−k2 cosets of C2 by choose different ∆st, we will use C2,s2 to denote them. B. Lattices and Nested Lattice Codes A n-dimensional lattice Λ is defined by all combination of a set of n real basis vectors g1, · · · ,g1 in Rn Λ = {l = G · i : i ∈ Zn} where Z = {0,±1,±2, · · ·} and G = [g1, · · · ,g1] is generator matrix. The nearest neighbor quantizer Q(·) is Q(x) = l ∈ Λ if‖x− l‖ ≤ ‖x− l′‖ The basic cell of Λ is V0 = {x : Q(x) = 0} The volume of Λ is V = ∫