Constructing optimal low rate convolutional codes

Constructing the optimal convolutional codes by analytic methods has been an open problem. Maximum free distance (MFD) encoders are found by exhaustive search on computers and by comparing with the upper bound on d_free. In this paper, on the basis of the Hamming weight representation by Hadamard matrices, we derive bounds for the components in the distance spectrum and necessary conditions for attaining these bounds. Then the optimal distance spectrum (ODS) codes that reach different level of these bounds are constructed for certain low code rates. The low code rate ODS codes can be used for so-called code-spreading CDMA