It's also straightforward to manipulate this algebra in python using the matrices

I=array([[1,0],[0,1]])  #I=eye(2)
X=array([[0,1],[1,0]])
Z=array([[1,0],[0,-1]])
Y=1j*X.dot(Z) # Y=array([[0,-1j],[1j,0]])

H=(X+Z)/sqrt(2)
x0=array([1,0])
x1=array([0,1])

Then use the kron ("kronecker direct product") function to define:

def p(A,B,C,D,E): return kron(A,kron(B,kron(C,kron(D,E))))

and thereby operators on the 5 Qbit space

M0=p(I,Z,X,X,Z)
M1=p(Z,I,Z,X,X)
M2=p(X,Z,I,Z,X)
M3=p(X,X,Z,I,Z)

I5=p(I,I,I,I,I)
H5=p(H,H,H,H,H)
X5=p(X,X,X,X,X)

and the encoded (0-bar and 1-bar) states

b0=(1/4.)*(I5+M0).dot(I5+M1).dot(I5+M2).dot(I5+M3).dot(p(x0,x0,x0,x0,x0))
b1=(1/4.)*(I5+M0).dot(I5+M1).dot(I5+M2).dot(I5+M3).dot(p(x1,x1,x1,x1,x1))

Matrix elements are, e.g.,

b0.dot(H5).dot(b1)


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Or in matlab, use the matrices

I=[1 0; 0 1];
X=[0 1; 1 0];
Z=[1 0; 0 -1];
Y=1j*X*Z;

H = [1 1; 1 -1] / sqrt(2)
x0=[1 0]';
x1=[0 1]';

Then use the kron ("kronecker direct product") function to define:

p = @(A,B,C,D,E) kron(A,kron(B,kron(C,kron(D,E))));

and thereby operators on the 5 Qbit space

M0=p(I,Z,X,X,Z);
M1=p(Z,I,Z,X,X);
M2=p(X,Z,I,Z,X);
M3=p(X,X,Z,I,Z);

I5=p(I,I,I,I,I);
H5=p(H,H,H,H,H);
X5=p(X,X,X,X,X);

and the encoded (0-bar and 1-bar) states

b0=(1/4)*(I5+M0)*(I5+M1)*(I5+M2)*(I5+M3)*p(x0,x0,x0,x0,x0);
b1=(1/4)*(I5+M0)*(I5+M1)*(I5+M2)*(I5+M3)*p(x1,x1,x1,x1,x1);

Matrix elements are, e.g.,

b0'*H5*b1