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Register a new userThe resolution of Nihos last conjecture concerning sequences, codes, and Boolean functions
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Added by
Daniel Katz
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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A new method is used to resolve a long-standing conjecture of Niho concerning the crosscorrelation spectrum of a pair of maximum length linear recursive sequences of length $2^{2 m}-1$ with relative decimation $d=2^{m+2}-3$, where $m$ is even. The result indicates that there are at most five distinct crosscorrelation values. Equivalently, the result indicates that there are at most five distinct values in the Walsh spectrum of the power permutation $f(x)=x^d$ over a finite field of order $2^{2 m}$ and at most five distinct nonzero weights in the cyclic code of length $2^{2 m}-1$ with two primitive nonzeros $alpha$ and $alpha^d$. The method used to obtain this result proves constraints on the number of roots that certain seventh degree polynomials can have on the unit circle of a finite field. The method also works when $m$ is odd, in which case the associated crosscorrelation and Walsh spectra have at most six distinct values.
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We give a new, purely coding-theoretic proof of Kochs criterion on the tetrad systems of Type II codes of length 24 using the theory of harmonic weight enumerators. This approach is inspired by Venkovs approach to the classification of the root systems of Type II lattices in R^{24}, and gives a new instance of the analogy between lattices and codes.
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