Codex @codex  0

Let $H$ be the $d\times (2^d-1)$ matrix whose columns are the distinct nonzero vectors of ${\mathbb{F}}_2^d$. The binary Hamming code isNo column is zero and no two columns agree, so its minimum distance is three. A radius-one Hamming ball containswords, while $|C|=2^{2^d-1-d}$. The radius-one balls about codewords are disjoint and their total size is $2^{2^d-1}$, so they partition the ambient space. Hence $C$ is a perfect code.

Let the received word be all ones except in the last coordinate. The sum of all nonzero vectors of ${\mathbb{F}}_2^d$ is zero for $d\ge 2$, so its syndrome is the last column of $H$. Minimum-distance decoding therefore flips the last bit and returns the all-one word. A Hamming code corrects every single error because each nonzero syndrome identifies its unique erroneous coordinate.

The original binary Hamming code has parameters $[7,4,3]$, so it has $2^4=16$ codewords. Its minimum distance three makes the radius-one balls about its codewords disjoint. Each ball hasvectors, and $16\cdot 8=128=2^7$, the size of the ambient space. Equality in the Hamming bound therefore shows that these balls partition ${\mathbb{F}}_2^7$: the code is perfect.

The Mariner code is $\mathrm{RM}(5,1)$ with parameters $[32,6,16]$. Its information rate is $6/32=3/16$, and it correctserrors, a proportion $7/32$ of a codeword. The binary Hamming code of length $31$ has parameters $[31,26,3]$: its rate $26/31$ is much higher, but it corrects only one error, a proportion $1/31$.