Constructing optimal convolutional codes

Constructing the optimal convolutional codes by analytic methods has been an open problem. Most encoders used in practice have been found by exhaustive search on computers. In this paper, on the basis of the Hamming weight presentation by Hadamard matrices and attainable upper bounds for free distance and the second smallest Hamming distance, we can construct some optimal convolutional codes in diffrent meanings. Specially very low rate optimal convolutional codes, which cannot be found by exhaustive search, are derived. These codes can be applied to so-called code-spread CDMA system. Index Terms--Convolutional code, Hadamard matrix, Attainable upper bounds, Code spread CDMA, Diophantine equation, integer programming.