Part 1
Here is the MATLAB code for solving the spin glass problem:

% set the number of spins in the glass
N = 100;

% random energies from a gaussian distribution with mean 0 and variance 1 
J = randn(1, N);

% print out the minimum possible energy
-sum(abs(J))

% construct a symmetric energy matrix A
D = diag(J);
D1 = [D(:,end) D(:,1:end-1)];
D2 = [D(end,:); D(1:end-1,:)];
A = 0.5 * (D1 + D2);

% allocate a symmetrix matrix for the spins
Z = sdpvar(N);

% constrain the spin matrix to be psd and have 1's on the diagonal
F = set(Z > 0) + set(diag(Z) == 1);

% solve the semidefinite program
solvesdp(F, trace(A*Z));

% print out the energy obtained
trace(A * double(Z))

ans =

  -78.2133

ans =

  -78.2133

Part 2
For this part, I did the same as above, except I wrote this ugly nested for loop to obtain the energy matrix A:

A = zeros(N);

for i = [1:N]
  col1 = mod(i - 1, S) + 1;
  row1 = (i - col1) / S + 1;

  down  = mod(row1, S) + 1;
  right = mod(col1, S) + 1;

  up    = mod(row1 + S - 2, S) + 1;
  left  = mod(col1 + S - 2, S) + 1;

  for j = [i + 1:N]
    col2 = mod(j - 1, S) + 1;
    row2 = (j - col2) / S + 1;
    
    if ((row1 == row2 && (col2 == right || col2 == left)) || (col1 == col2 && (row2 == down || row2 == up)))
      r = 0.5 * randn(1,1);
      A(i, j) = r;
      A(j, i) = r;
    end
  end
end