Timeline for Maximum probability of a set of vectors from $ \mathbb{F}_2^n $ being linearly independent
Current License: CC BY-SA 4.0
7 events
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| Jan 12, 2021 at 8:07 | comment | added | aleph | @OfirGorodetsky Thanks for the references, I wasn't aware of the second one. | |
| Jan 11, 2021 at 22:19 | comment | added | Ofir Gorodetsky | Do not have an answer for fixed n, but there is asymptotic sense in which the distribution doesn't matter unless it is very localized, see 1) Kahn J. and Komlós J. (2001), Singularity Probabilities for Random Matrices over Finite Fields, Combinatorics, Probability and Computing, 10(2), 137-157 and 2) Vinh L. A. (2012), Singular matrices with restricted rows in vector spaces over finite fields, Discrete Mathematics, 312(2), 413-418. | |
| Jan 11, 2021 at 21:57 | history | edited | aleph | CC BY-SA 4.0 |
added 232 characters in body
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| Jan 11, 2021 at 21:50 | comment | added | aleph | @LSpice To your first question - I don't know :) To your second question - yes, selecting some vector twice would mean that the condition of linear independence is violated. I'll edit my question to clarify. | |
| Jan 11, 2021 at 21:38 | comment | added | LSpice | Also, just to be clear, if a vector is ever selected twice, you count that as a failure, right? (That is, you are really looking at the indexed family of selections, not just the set of selected vectors?) | |
| Jan 11, 2021 at 21:38 | comment | added | LSpice | I know it's an obvious and silly question, but: obviously the 'best' $P$ will assign $0$ weight to $0$. (Otherwise, just re-assign that weight to any non-$0$ vector to increase the "linear independence probability".) Suppose that it assigns different weights $p$ and $q$ to two non-$0$ vectors $v$ and $w$. Can it ever happen that the distribution that assigns the average weight $(p + q)/2$ to both $v$ and $w$, and is otherwise unchanged, has a lower "linear independence probability"? | |
| Jan 11, 2021 at 21:00 | history | asked | aleph | CC BY-SA 4.0 |