Xrfgtjtk

  Nota: Para cidade estadunidense no Texas, com nome semelhante, veja Edinburg. Coordenadas: 55° 57' 00" N, 3° 13' 12" W Edimburgo Edinburgh Geografia Área Total 264 km² ( posição) Categoria Council Area, Cidade Demografia População Total (2011 [ 1 ] ) 495.360 habitantes (2 posição) Densidade 4,776 hab./km² Política Nome do Conselho City of Edinburgh Council Site http://www.edinburgh.gov.uk ...

up vote 1 down vote favorite Note: We are talking about binary codes. Definition 1 : For integers $1 ≤ d ≤ n$ , a code $C$ is an $(n, d)$ -code if it has length $n$ and minimum distance $d_H (C) ≥ d$ . An $(n, M, d)$ -code is an $(n,d)$ -code of size $M$ . Definition 2 : For integers $1 ≤ d ≤ n$ , let $A(n,d)$ be the largest $M$ such that there exists an $(n, M, d)$ -code. An $(n, d)$ -code is optimal if has size $A(n, d)$ . Prove that for any integer $n ≥ 2$ and $1 ≤ d ≤ n$ , we have $$A(n, d) ≤ 2A(n − 1, d)$$ . What I did: It suffices to prove that given $C$ , a $(n,d)$ -code, we can find $C'$ , a $(n-1,d)$ -code, such that $|C| leq 2|C'|.$ Next what I think is that we have to truncate $C$ by deleting the last bit of every codeword. This will make $C'$ a code of length $n-1$ but doing so will affect t...