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Improving the Gilbert-Varshamov bound for $q$-ary codes
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05) (2005).

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Given positive integers $q$, $n$ and $d$, denote by $A_q(n,d)$ the maximum size of a $q$-ary code of length $n$ and minimum distance $d$. The famous Gilbert-Varshamov bound asserts that $A_q(n,d+1) \geq q^n / V_q(n,d)$, where $V_q(n,d)=\sum_{i=0}^d \binom{n}{i}(q-1)^i$ is the volume of a $q$-ary sphere of radius $d$. Extending a recent work of Jiang and Vardy on binary codes, we show that for any positive constant $\alpha$ less than $(q-1)/q$ there is a positive constant $c$ such that for $d \leq \alpha n, A_q(n,d+1) \geq c \frac{q^n}{ V_q(n,d)}n$. This confirms a conjecture by Jiang and Vardy. 
@article{DMTCS_2005_special_250_a65,
     author = {Vu, Van H. and Wu, Lei},
     title = {Improving the {Gilbert-Varshamov} bound for $q$-ary codes},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)},
     year = {2005},
     doi = {10.46298/dmtcs.3456},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3456/}
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Vu, Van H.; Wu, Lei. Improving the Gilbert-Varshamov bound for $q$-ary codes. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05) (2005). doi : 10.46298/dmtcs.3456. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3456/

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