While ho =h1,hol =+1.0,entropy H(ylso +s1)= The PMF propagated from Mit to Xi.j is given by H(ylso,s1).And while ho =-h1,hol +1.0,entropy H(ylso +s1)is given by PM.L→X（(Xi) 0w+-2e票》计 A∑ Π Px→M(X ·(18) x41j≠j,(X,,M.)e9 3am2)+m(22加 Ij',Lxigsi)]Ps-M(sit)I(j.L.xij.si.). 2σ2 The PMF propagated from Ci.!to Xi.j is given by B.PMF from Variable Node to Check Node in Algorithm 1 PC1→X,（xi） The PMF propagated from Si.j to Yi is given by A∑[Π Px→C(Xr小 (19) the Ist iteration, =x4,11≠j,(X,,C,l)eg Ps,→y(si)= 1/s4, PMy→S,(si0.w… 6⊕(xr,CEgx,rll小 (11) The PMF propagated from Si.j to Mi.j is given by For client subset Vi.let B The PMF propagated from Ei.j to Xi.j is given by ∫1/s4 the Ist iteration. PS,→M,(si)= P一X,（X) Py,→S(si,o.w.… =A·∑[Pz,-→8，(2）i(Bx-z小 (20) (12) The PMF propagated from Xi.j to Mi.is given by xi,j PX→M(Xi）= The PMF propagated from Ei.j to Zij is given by 1/21 the Ist iteration, P841→Z4,（2） =A.∑[Pxw→8，(x)6(Bgx-小 (21) A.Π PC→X（0.w. 心24， (XiCL)EG (13) D.Calculation of Covariance Matrix Ti.j The PMF propagated from Xi.j to Ci.is given by Random variables in vector Real(yj)follow multivariate PX,1→C(Xij）= normal distribution.On condition of random variable vector 1/2w the 1st iteration, sij,we can calculate the covariance matrix Ti;of random variables in vector Real(y;)as follows A. Π Pc→X,(X） l≠，(X,C,)e T:j=>(Var(Re(s))Real(h)Real(h)T+ 0.w. IEV/V Π PM.→X,(X） (Xi.j.M)EG Var(Im(s))Imag()Imag(h)T+ (14) Cov(Re(sv.j),Im(svL.j)) The PMF propagated from Xi.i to i.j is given by (Real(h)Imag(h)T+Imag(h)Real(h)T))+ PX→2，(x)=A.ΠPCu→X,(xH ∑(Cov(Re(s,Re(sr》 (X4,C4,)e9 Vey/},reV,l≠ (15) ΠPMA→X,x) (Real(hi )Real(hi)T +Real(h)Real(T)+(22) (Xi.j,Mi.1)EG Cov(m(su,,Im(sv,》 The PMF propagated from Zij to ij is given by (Imag(h)Imag(h+Imag(h)Imag(hT)+ Pz4→84，(a）=2-1M (16) Cov(Re(svj),Im(svj)). C.PMF from Check Node to Variable Node in Algorithm I (Real()Imag(h)T+Imag()Real(T)+ The PMF propagated from Mi.j to Si.j is given by Cov(Im(s,),Re(sr,)片 PM→S(si) (Imag(h)Real(hT +Real(h)Imag(h)T))+ A∑ Ⅱ[Px→aM,x) diag(…,o2,…)）为 (17) =5,(X,,M)e9 where Var()denotes the variance of random variable, I0,i,x,s】 Cov()denotes the covariance of random variables,both of which are obtained based on PMF Ps(s). where I(j',j,xi,sij)is the indicator function to ensure the diag()denotes diagonal matrix.And operation Imag(): j'th codeword bits vector is consistent with the j-thc is defined as Imag([oo) symbol vetor si.j. 【-m(ao),Re(ao,…,-Im(awl-1),Re(a4wl-1JWhile h0 = h1, |h0| = +1.0, entropy H(y|s0 + s1) = H(y|s0, s1). And while h0 = −h1, |h0| = +1.0, entropy H(y|s0 + s1) is given by H(y|s0 + s1) = 1 2 H{ 1 √ 2πσ ￾ exp(−y 2 2σ 2 )
}+ 1 2 H{ 1 2 √ 2πσ ￾ exp(−(y + 2)2 2σ 2 ) + exp(−(y − 2)2 2σ 2 )
}. B. PMF from Variable Node to Check Node in Algorithm 1 The PMF propagated from Si,j to Yj is given by PSi,j→Yj (si,j ) = ( 1/|S||Vi| , the 1st iteration, PMi,j→Si,j (si,j ), o.w.. (11) The PMF propagated from Si,j to Mi,j is given by PSi,j→Mi,j (si,j ) = ( 1/|S||Vi| , the 1st iteration, PYj→Si,j (si,j ), o.w.. (12) The PMF propagated from Xi,j to Mi,l is given by PXi,j→Mi,l (xi,j ) =    1/2 |Vi| , the 1st iteration, A · Y (Xi,j ,Ci,l′ )∈G PCi,l′→Xi,j (xi,j ), o.w.. (13) The PMF propagated from Xi,j to Ci,l is given by PXi,j→Ci,l (xi,j ) =    1/2 |Vi| , the 1st iteration, A · Y l6=l ′ ,(Xi,j ,Ci,l′ )∈G PCi,l′→Xi,j (xi,j ) · Y (Xi,j ,Mi,l′ )∈G PMi,l′→Xi,j (xi,j ) , o.w.. (14) The PMF propagated from Xi,j to Σi,j is given by PXi,j→Σi,j (xi,j ) = A · Y (Xi,j ,Ci,l)∈G PCi,l→Xi,j (xi,j )· Y (Xi,j ,Mi,l)∈G PMi,l→Xi,j (xi,j ). (15) The PMF propagated from Zi,j to Σi,j is given by PZi,j→Σi,j (zi,j ) = 2−|Vi| . (16) C. PMF from Check Node to Variable Node in Algorithm 1 The PMF propagated from Mi,j to Si,j is given by PMi,j→Si,j (si,j ) = A · X ∼si,j Y (Xi,j′ ,Mi,j )∈G PXi,j′→Mi,j (xi,j′ )· I(j ′ , j, xi,j′ , si,j ) , (17) where I(j ′ , j, xi,j′ , si,j ) is the indicator function to ensure the j ′ -th codeword bits vector xi,j′ is consistent with the j-th symbol vetor si,j . The PMF propagated from Mi,l to Xi,j is given by PMi,l→Xi,j (xi,j ) = A · X ∼xi,j Y j6=j ′ ,(Xi,j′ ,Mi,l)∈G PXi,j′→Mi,l (xi,j′ )· I(j ′ , l, xi,j′ , si,l) PSi,l→Mi,l (si,l)I(j, l, xi,j , si,l). (18) The PMF propagated from Ci,l to Xi,j is given by PCi,l→Xi,j (xi,j ) = A · X ∼xi,j Y j6=j ′ ,(Xi,j′ ,Ci,l)∈G PXi,j′→Ci,l (xi,j′ )· δ(k ⊕(Xi,j′ ,Ci,l)∈G xi,j′k1) . (19) For client subset Vi , let Bi = [βˆ vi0 , · · · , βˆ vi |Vi |−1 ] T . The PMF propagated from Σi,j to Xi,j is given by PΣi,j→Xi,j (xi,j ) =A · X ∼xi,j PZi,j→Σi,j (zi,j )δ(B T i xi,j − zi,j ) . (20) The PMF propagated from Σi,j to Zi,j is given by PΣi,j→Zi,j (zi,j ) =A · X ∼zi,j PXi,j→Σi,j (xi,j )δ(B T i xi,j − zi,j ) . (21) D. Calculation of Covariance Matrix Γi,j Random variables in vector Real(yj ) follow multivariate normal distribution. On condition of random variable vector si,j , we can calculate the covariance matrix Γi,j of random variables in vector Real(yj ) as follows Γi,j = X vl∈V /Vi ￾ Var(Re(svl,j ))Real(h j vl )Real(h j vl ) T + Var(Im(svl,j ))Imag(h j vl )Imag(h j vl ) T + Cov(Re(svl,j ),Im(svl,j ))· (Real(h j vl )Imag(h j vl ) T + Imag(h j vl )Real(h j vl ) T )
+ X Vi ′∈V/{Vi} X vl,vl ′∈Vi ′ ,l6=l ′ ￾ Cov(Re(svl,j ), Re(svl ′ ,j ))· (Real(h j vl )Real(h j vl ′ ) T + Real(h j vl ′ )Real(h j vl ) T )+ Cov(Im(svl,j ),Im(svl ′ ,j ))· (Imag(h j vl )Imag(h j vl ′ ) T + Imag(h j vl ′ )Imag(h j vl ) T )+ Cov(Re(svl,j ),Im(svl ′ ,j ))· (Real(h j vl )Imag(h j vl ′ ) T + Imag(h j vl ′ )Real(h j vl ) T )+ Cov(Im(svl,j ), Re(svl ′ ,j ))· (Imag(h j vl )Real(h j vl ′ ) T + Real(h j vl ′ )Imag(h j vl ) T )
+ diag(· · · , σ2 , · · ·), (22) where Var(·) denotes the variance of random variable, Cov(·) denotes the covariance of random variables, both of which are obtained based on PMF PSi ′,j→Yj (si ′ ,j ). diag(·) denotes diagonal matrix. And operation Imag(·) : C |U| → R 2|U| is defined as Imag([α0, · · · , α|U|−1] T ) , [−Im(α0), Re(α0), · · · , −Im(α|U|−1), Re(α|U|−1)]T .