Uniquely Decodable Codes And Decoder for Overloaded Synchronous CDMA Systems

A decoding method for synchronous code division multiple access (CDMA) comprising:

• for S1, given that Y=[y1,y2]T=S1X and {circumflex over (X)} be the decoded data:
• passing y1 to a quaternary Pulse amplitude modulated (PAM) decoder with constellation of {±1, ±3} to generate an output of z, which shows the number of +1s and −1s in {circumflex over (X)};
• If z=+3 or z=−3, then, stop;
• otherwise, from y2 and z:
• if z=+1:
• passing y2 to a ternary PAM decoder with constellation of {0, ±2}, to determine which user sent said −1;
• if z=−1:
• passing y2 to a ternary PAM decoder with constellation of {0, ±2} to determine which user sent said +1;
• for Sk, k≧2 passing y1 to a (2k+1)-ary PAM decoder with a constellation of {±1, ±3, ±(2k+1−1)} to generate an output of z, which shows the number of +1s and −1s in {circumflex over (X)};
• If z is +(2k+1−1) or −(2k+1−1), then the process is terminated and {circumflex over (X)} is composed purely of +1s or −1s, respectively;
• otherwise, passing y2 to a (2k+1−|z|)-ary PAM decoder with a constellation of {0, ±2, ±(2k+1−1−|z|)} to generate an output;
• combining said output and information about the total number of −1s and +1s in {circumflex over (X)}, it is possible to decipher the 2kth entry of {circumflex over (X)}. Moreover, one may determine the number of +1s and −1s in the first and last 2k−1 entries of {circumflex over (X)}, denoted by nl and nr, respectively.
• Applying the decoder of Sk−1 with inputs of
• Y
• l
• =
• [
• 2
• k
• -
• 1
• -
• 1
• -
• 2
• 
• n
• l
• y
• 3
• ⋮
• y
• 2
• k
• -
• 1
• +
• 1
• ]
• 
• 
• and
• 
• 
• Y
• r
• =
• [
• 2
• k
• +
• 1
• -
• 1
• -
• 2
• 
• n
• r
• y
• 2
• k
• -
• 1
• +
• 1
• ⋮
• y
• 2
• k
• ]
• (
• 3
• )
• to identify the first and last (2k−1) entries of {circumflex over (X)}.