You are an expert in writing code in many different languages. Your goal is to perform code completion for the following code, keeping in mind the rest of the code and the file meta data.
    Metadata details:
    filename: DCA_polar_decoder.m
    path of file: ./repos/polar-3gpp-matlab/components
    
    the code of the file until where you have to start completion: function a_hat = DCA_polar_decoder(e_tilde, crc_polynomial_pattern, crc_interleaver_pattern, info_bit_pattern, rate_matching_pattern, mode, L, min_sum, P2)
% DCA_POLAR_DECODER Distributed-CRC-Aided (DCA) polar decoder.
%   a_hat = DCA_POLAR_DECODER(e_tilde, crc_polynomial_pattern, crc_interleaver_pattern, info_bit_pattern, rate_matching_pattern, mode, L, min_sum, P2) 
%   decodes the encoded LLR sequence e_tilde, in ord
    
function calls to perform the XOR, g and f functions
    % necessary to obtain the corresponding LLR
    update_llr(i,1);
    
    if info_bit_pattern(i) == 0 % Frozen bit
        PM = phi(PM, llrs(i,1,:), 0); 
    else % Information or CRC bit
        % Double the list size, using 0-valued bits for the first half and 1-valued bits for the other half
        PM = cat(3,phi(PM, llrs(i,1,:), 0), phi(PM, llrs(i,1,:), 1));
        llrs = cat(3,llrs,llrs);
        bits = cat(3,bits,bits);
        bits(i,1,1:L_prime) = 0;
        bits(i,1,L_prime+1:2*L_prime) = 1;
        bits_updated(i,1) = true;
        
        % We use the interleaved CRC generator matrix to update the CRC 
        % check sums whenever an information or CRC bit adopts a value of
        % 1.
        crc_checksums = cat(3,crc_checksums,mod(crc_checksums+repmat(G_P3(i2,:)',[1 1 L_prime]),2));
        % We need to keep track of whether any of the checks associated 
        % with the previous CRC bits have failed.
        crc_okay = cat(3,crc_okay,crc_okay);
              
        % If the list size has grown above L, then we need to find and keep only the best L entries in the list
        L_prime = size(bits,3);        
        if L_prime > L
            [~,max_indices] = sort(PM,3);
            PM = PM(:,:,max_indices(1:L));
            bits = bits(:,:,max_indices(1:L));
            llrs = llrs(:,:,max_indices(1:L));
            crc_checksums = crc_checksums(:,:,max_indices(1:L));
            crc_okay = crc_okay(:,:,max_indices(1:L));
            L_prime = L;
        end