Maximum code words

Question

i am new to coding theory. i am trying to understand some of the basics by solving a few questions. I came across this one. Assuming $\\C$ is a binary (not necessarily linear) code of length $\\n$ and minimum hamming distance $\\d$. Atmost how many codewords can $\\C$ have? Give an example to the extremal case. I am stuck at the example part. i think using the fact that $\left|C\right|\times \sum _{r=0}^e^nC_r\le 2^n$ - am i right? I dont know what to do about the example part.

Answer 1

The sphere packing bound says that the number of codewords in a code with length $n$ and distance d over an alphabet of size $q$ is at most $\frac{q^n}{\sum_{k=0}^{\lfloor \frac{d-1}{2} \rfloor} \binom{n}{k} (q-1)^k}$.

Codes which attain equality are said to be perfect. Examples of perfect codes which are non-trivial are hamming and golay codes (in fact, the Hamming code, Binary Golay and Ternary Golay are the only parameters you can get for non-trivial perfect codes over a finite field).