Perfect code

Perfect codes

Codes that attain the Hamming bound are called perfect codes. Examples include codes that have only one codeword, and codes that are the whole of ${\displaystyle {\mathcal {A}}_{q}^{n}}$. Another example is given by the repeat codes, where each symbol of the message is repeated an odd fixed number of times to obtain a codeword where q = 2. All of these examples are often called the trivial perfect codes. In 1973, Tietäväinen proved that any non-trivial perfect code over a prime-power alphabet has the parameters of a Hamming code or a Golay code.