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Let $C$ be a linear code with minimum distance $2k$. I want to show that there is a coset of $C$ that contains at least two vectors of weight $k$.

Firstly, it holds that the minimum distance of the code is equal to the lowest non-zero weight of a codeword. So this means that the weights of the codewords are greater than these of the elements of the coset of $C$, right?

Solving the system $Hx=0$, where $H$ is the parity matrix, we can find the coset of $C$, right?

But in this case we don't have a generator matrix of the dual code. How can we get information about the parity matrix?

Or how else can we get information about the coset of $C$?

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Let $v$ be any vector of weight $k$. Then all vectors in $C+v$ have weight at least $k$. Furthermore, if $w \in C$ is any vector of weight exactly $2k$ (such a vector must exist since the minimal weight of $C$ is $2k$) and $v$ is a "subset" of $w$ then $v,w+v \in C+v$ are two vectors of weight exactly $k$.

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  • $\begingroup$ There are far too many comments here! @Evinda and Yuval, please remove obsolete comments. $\endgroup$ – Raphael Feb 24 '16 at 20:54
  • $\begingroup$ @YuvalFilmus Does it hold that a coset of a code $C$ is $C$ itself? $\endgroup$ – Evinda Mar 1 '16 at 20:54
  • $\begingroup$ @YuvalFilmus Also does the proposition of my initial post hold for all codes or anly for the binary ones? $\endgroup$ – Evinda Mar 1 '16 at 21:03
  • $\begingroup$ I believe you have in your possession all the tools to answer these questions on your own. $\endgroup$ – Yuval Filmus Mar 1 '16 at 23:02

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