Measures of pseudorandomness showing binary sequence with small correlation measure of order k up to a sufficiently large k cannot have a large arithmetic correlation applying the result to several classes of sequences including Legendre sequences defined with polynomials

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Measures of pseudorandomness showing binary sequence with small correlation measure of order k up to a sufficiently large k cannot have a large arithmetic correlation applying the result to several classes of sequences including Legendre sequences defined with polynomials.

Measures of Pseudorandomness: Arithmetic Autocorrelation and Correlation Measure -

https://www.semanticscholar.org/paper/Measures-of-Pseudorandomness%3A-Arithmetic-and-Hofer-M%C3%A9rai/5d721da76d6867f97d309d681669f73f0d0c4299 

 

Consider a simplified example to illustrate the phenomenon described in the paper.

Imagine we have a binary sequence: 0101010101010101. This sequence alternates between 0s and 1s. If we calculate the correlation measure of order 1 for this sequence, it will be small because adjacent elements are different.

Now, let's calculate the arithmetic autocorrelation for the same sequence. The arithmetic autocorrelation measures the correlation between the elements at different positions in the sequence. In this case, if we compare the elements at positions 1 and 2, they are the same (both 0s). Similarly, if we compare the elements at positions 2 and 3, they are also the same (both 1s). This pattern continues throughout the sequence.

Due to this strong pattern of repeated elements, the arithmetic autocorrelation for this sequence will be large.

According to the paper's findings, if a binary sequence has a small correlation measure of order k for all sufficiently large values of k, it cannot have a large arithmetic correlation. In our example, since the correlation measure of order 1 is small, it implies that the arithmetic autocorrelation cannot be large.

This phenomenon is applicable to various classes of sequences, including Legendre sequences defined with polynomials. The paper explores these classes of sequences and demonstrates how the measures of pseudorandomness can help analyze their properties and determine the presence or absence of patterns.

 

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