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Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that can give bounds or estimates for the probability that the kernel of $M$ contains at least one short non-zero vector $v\in \mathbb{Z}^n $?

By short I mean with small $L_2$ norm. Although I am interested in a general theory, my specific interest is in vectors no longer than $\sqrt{n}$.

Added I would also be interested in any ideas for the $L_1$ norm as well. Specifically if the $L_1$ norm is bounded by a small constant.

Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that can give bounds or estimates for the probability that the kernel of $M$ contains at least one short non-zero vector $v\in \mathbb{Z}^n $?

By short I mean with small $L_2$ norm. Although I am interested in a general theory, my specific interest is in vectors no longer than $\sqrt{n}$.

Added I would also be interested in any ideas for the $L_1$ norm as well. Specifically if the $L_1$ norm is bounded by $n$ for example.

Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that can give bounds or estimates for the probability that the kernel of $M$ contains at least one short non-zero vector $v\in \mathbb{Z}^n $?

By short I mean with small $L_2$ norm. Although I am interested in a general theory, my specific interest is in vectors no longer than $\sqrt{n}$.

Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that can give bounds or estimates for the probability that the kernel of $M$ contains at least one short non-zero vector $v\in \mathbb{Z}^n $?

By short I mean with small $L_2$ norm. Although I am interested in a general theory, my specific interest is in vectors no longer than $\sqrt{n}$.

Added I would also be interested in any ideas for the $L_1$ norm as well. Specifically if the $L_1$ norm is bounded by a small constant.