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15 votes
2 answers
3k views

How to compute the rank of a matrix?

Okay, that's a misleading title. This is a somewhat subtler problem than undergraduate linear algebra, although I suspect there's still an easy answer. But I couldn't resist :D. Here's the actual ...
Harrison Brown's user avatar
5 votes
2 answers
680 views

Finding the solution to b = Ax that minimizes the Hamming weight (everything over the field F_2).

Is there an efficient algorithm for finding the solution $x$ of $b = Ax$ that minimizes the Hamming weight of $x$, where $A$ is a nxm-matrix over the field $\mathbb{F}_2$ ("integer matrix modulo 2")...
David's user avatar
  • 141
5 votes
0 answers
240 views

Complexity of approximating the size of the range of a matrix

Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define: $$S_M = |\{Mx : x \in \{-1,1\}^n\}.$$ It is NP-hard to compute $S_M$ exactly I believe by applying the ...
Simd's user avatar
  • 3,377
0 votes
0 answers
104 views

Efficient Algorithm to Find Subset of Vectors Over $\mathbb{F}_q$ Living in Low Dimensional Subspace

Let $q$ be a fixed prime, $P, Q$ be polynomials with $\mathrm{deg}(Q) < \mathrm{deg}(P)$ and $h = O(\log n)$. Let $S$ be a subset of $\mathbb{F}_q^n$ of size $P(n)$ such that there exists a subset ...
cha21's user avatar
  • 328