Introduction to Coding Theory
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We study a problem motivated by a question related to quantum error-correcting codes. Combinatorially, it involves the graph parameter $f(G)=\min\{|A|+|\{x\in V\setminus A:d_A(x)$ is $\text{odd}\}|:A\neq\emptyset\}$, where $V$ is the vertex set of $G$ and $d_A(x)$ is the number of neighbors of $x$ in $A$. We give asymptotically tight estimates of $f$ for the random graph $G_{n,p}$ when $p$ is constant. Also, if $f(n)=\max\{f(G):\,|V(G)|=n\}$, then we show that $f(n)\leq(0.382+o(1))n$.