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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    $\begingroup$ 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 Petrov Nov 16 '16 at 16:43
  • $\begingroup$ Addressed here: mathoverflow.net/questions/24864/almost-orthogonal-vectors $\endgroup$ – Ryan O'Donnell Nov 16 '16 at 17:35
  • $\begingroup$ @FedorPetrov Thank you. Can you point me some references? $\endgroup$ – lchen Nov 16 '16 at 18:17
  • $\begingroup$ say, J.H. van Lint (1992). Introduction to Coding Theory. $\endgroup$ – Fedor Petrov Nov 16 '16 at 20:31

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