“Dual of BCH is an independent source”的意思、由来-开放百科全书

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Dual of BCH is an independent source

释义

Lemma
Proof of Lemma
Corollary
Proof of Corollary
References

A certain family of BCH codes have a particularly useful property, which is that

treated as linear operators, their dual operators turns their input into an -wise independent source. That is, the set of vectors from the input vector space are mapped to an -wise independent source. The proof of this fact below as the following Lemma and Corollary is useful in derandomizing the algorithm for a -approximation to MAXEkSAT.

Lemma

Let be a linear code such that has distance greater than . Then is an -wise independent source.

Proof of Lemma

It is sufficient to show that given any matrix M, where k is greater than or equal to l, such that the rank of M is l, for all , takes every value in the same number of times.

Since M has rank l, we can write M as two matrices of the same size, and , where has rank equal to l. This means that can be rewritten as for some and .

If we consider M written with respect to a basis where the first l rows are the identity matrix, then has zeros wherever has nonzero rows, and has zeros wherever has nonzero rows.

Now any value y, where , can be written as for some vectors .