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I've been reading about perfect codes and working on various exercises to get a better understanding about these types of codes. I came across an interesting statement that I am having trouble showing.

A coset of a linear perfect code is also a perfect code.

Can anyone help? Thanks in advance!

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    the second edition of Lattices and Codes by Wolfgang Ebeling, on page 66, deines a perfect code in a way you might like better. About five pages on this.2012-09-29

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Let $C$ be a linear code, let $v+C$ be a coset. Given two elements $x,y$ in the coset, we have $x=v+a$, $y=v+b$ for some $a,b$ in $C$. Can you show that the distance between $x$ and $y$ is the same as the distance between $a$ and $b$? Can you see how to apply that to your question?

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    Ok, I think I understand how to use the distance property. The book also says that a perfect code $C$ is one for which $\mathbb{F}_{q}^{n}$ is partitioned into disjoint spheres of radius $t$ centered about each of the codewords in $C$. Since all distances in $v + C$ are the same as distances in $C$, it would also follow that $\mathbb{F}_{q}^{n}$ would be partitioned into disjoint spheres of radius $t$ centered about the elements ${v + c}$, where $c$ runs over all of $C$.2012-09-30