What is the maximum number of mutually orthogonal $n$-bit sequences can we construct? And how to construct them? A trivial example is using the Hadamard matrix, but we can only build $n$ orthogonal $n$-bit sequences. Can we do better? If we relax the orthogonality such that we require sequences to have low correlation, what is the maximum number of $n$-bit sequences with low correlation? And how to construct them?
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2$\begingroup$ There are at most $n$ mutually orthogonal vectors in $\mathbb{R}^n$, how can we hope to do better? For low correlation, this is about finding large binary codes with prescribed code distance, there are many estimates from both sides in this problem. $\endgroup$– Fedor PetrovCommented Nov 16, 2016 at 16:43
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$\begingroup$ Addressed here: mathoverflow.net/questions/24864/almost-orthogonal-vectors $\endgroup$– Ryan O'DonnellCommented Nov 16, 2016 at 17:35
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$\begingroup$ @FedorPetrov Thank you. Can you point me some references? $\endgroup$– lchenCommented Nov 16, 2016 at 18:17
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$\begingroup$ say, J.H. van Lint (1992). Introduction to Coding Theory. $\endgroup$– Fedor PetrovCommented Nov 16, 2016 at 20:31
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