Linear complexity in coding theory
The linear complexity of sequences is defined and its main properties reviewed. The linear complexity of periodic sequences is examined in detail and an extensive list of its properties is formulated. The discrete Fourier transform (DFT) of a finite sequence is then connected to the linear complexity of a periodic sequence by Blahut's theorem. Cyclic codes are given a DFT formulation that relates their minimum distance to the least linear complexity among certain periodic sequences. To illustrate the power of this approach, a slight generalization of the Hartmann-Tzeng lower bound on minimum distance is proved, as well as the Bose-Chaudhuri-Hocquenghem lower bound. Other applications of linear complexity in the theory of cyclic codes are indicated.
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