# Lee metric, Gilbert-Varshamov and Hamming bounds for larger relative distance ranges

Gardy and Sole 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.

• Closed for simultaneous cross-posting on MO. If you don't get an answer on MO after a few days, flag the post and we will reopen it. – Kaveh Jul 16 '11 at 22:37
• @Kaveh ok Thankyou – v s Jul 17 '11 at 0:13