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A Construction of Odd Length Generators for Optimal Families of Perfect Sequences
Song, Min Kyu,Song, Hong-Yeop IEEE 2018 IEEE transactions on information theory Vol.64 No.4
<P>In this paper, we give a construction of optimal families of <TEX>$N$</TEX>-ary perfect sequences of period <TEX>$N^{2}$</TEX>, where <TEX>$N$</TEX> is a positive odd integer. For this, we re-define perfect generators and optimal generators of any length <TEX>$N$</TEX> which were originally defined only for odd prime lengths by Park, Song, Kim, and Golomb in 2016, but investigate the necessary and sufficient condition for these generators for arbitrary length <TEX>$N$</TEX>. Based on this, we propose a construction of odd length optimal generators by using odd prime length optimal generators. For a fixed odd integer <TEX>$N$</TEX> and its odd prime factor <TEX>$p$</TEX>, the proposed construction guarantees at least <TEX>$(N/p)^{p-1}\phi (N/p)\phi (p)\phi (p-1)/\phi (N)^{2}$</TEX> inequivalent optimal generators of length <TEX>$N$</TEX> in the sense of constant multiples, cyclic shifts, and/or decimations. Here, <TEX>$\phi (\cdot )$</TEX> is Euler’s totient function. From an optimal generator one can construct lots of different <TEX>$N$</TEX>-ary optimal families of period <TEX>$N^{2}$</TEX>, all of which contain <TEX>$p_{\text {min}}-1$</TEX> perfect sequences, where <TEX>$p_{\text {min}}$</TEX> is the least positive prime factor of <TEX>$N$</TEX>.</P>