<p><a href="ftp://ftp.cs.brown.edu/pub/techreports/91/cs91-29.pdf" rel="nofollow noreferrer">Gardy and Sole</a> provide a Gilbert-Varshamov lower bound and a Hamming upper bound for the Lee metric when the distance between codewords is smaller than the length of the code (captured by <span class="math-container">$r = \delta n$</span> where <span class="math-container">$0 \le \delta \le 0.37$</span> in the paper).</p>

<p>Is there a reference which provides the corresponding lower/upper bound for ranges above this? At least is there a lower bound (preferably as powerful as the Gilbert-Varshamov bound) and a upper bound(preferably tighter than the Hamming bound) for the case <span class="math-container">$\frac{2 \delta n + 1}{n} = \frac{n + 1}{n}$</span>, that is <span class="math-container">$\delta = \frac{1}{2}$</span>?</p>

<p><em>This is a smaller version of the cross-posting from <a href="https://mathoverflow.net/questions/70524/lee-metric-constructive-and-asymptotic-bounds">MO</a>.</em></p>

CC BY-SA 3.0
2011-07-16T22:37:54.363Z
Closed for _simultaneous_ [cross-posting on MO](http://mathoverflow.net/questions/70524/lee-metric-constructive-and-asymptotic-bounds). If you don't get an answer on MO after a few days, flag the post and we will reopen it.
Kaveh
CC BY-SA 3.0
2011-07-17T00:13:23.247Z
@Kaveh ok Thankyou
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<coding-theory>
Lee metric, Gilbert-Varshamov and Hamming bounds for larger relative distance ranges
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# Lee metric, Gilbert-Varshamov and Hamming bounds for larger relative distance ranges

(Asked by: username_0 on 2011-07-16)

[Gardy and Sole](ftp://ftp.cs.brown.edu/pub/techreports/91/cs91-29.pdf) provide a Gilbert-Varshamov lower bound and a Hamming upper bound for the Lee metric when the distance between codewords is smaller than the length of the code (captured by $r = \delta n$ where $0 \le \delta \le 0.37$ in the paper).

Is there a reference which provides the corresponding lower/upper bound for ranges above this? At least is there a lower bound (preferably as powerful as the Gilbert-Varshamov bound) and a upper bound(preferably tighter than the Hamming bound) for the case $\frac{2 \delta n + 1}{n} = \frac{n + 1}{n}$, that is $\delta = \frac{1}{2}$?

*This is a smaller version of the cross-posting from [MO](https://mathoverflow.net/questions/70524/lee-metric-constructive-and-asymptotic-bounds).*

**Tags:** coding-theory
> **Comments:**
> **username_1**: Closed for _simultaneous_ [cross-posting on MO](http://mathoverflow.net/questions/70524/lee-metric-constructive-and-asymptotic-bounds). If you don't get an answer on MO after a few days, flag the post and we will reopen it.  
> **username_0**: @Kaveh ok Thankyou  

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