# math

## Error Correcting Codes Notes: Gilbert Bound

Table of Contents Review BSC Singleton Bound Hamming Bound Gilbert Bound Error Correcting Measures Basic Arithmetic GCD Review $Q^n$. $|Q|=q$. $C \subseteq Q^n$. $d_H(x,y).$ $$d_{\min}(C) = \min_{x,y \in C, x \neq y} d_H(x,y)$$ $wt(x)$ = # non-zeros in x. $S_r(x) = \{ y \in Q^n \mid d_H(x,y) \le r \}$ $$|S_r(x)| = \sum_{i=0}^r \begin{pmatrix}n\\i\end{pmatrix}(q-1)^i$$ BSC BSC(p)$_{0 < p < \frac{1}{2}}$. $R$ is an alphabet.