<p>The exact answer is unknown in general. One standard upper bound for $q-$ary codes is the Singleton bound, which gives
$$|C|\leq q^{n-d+1},$$ and codes meeting this bound are called MDS.</p>

<p>A lower bound is the Gilbert-Varshamov bound, given by</p>

<p>$$|C|\geq \frac{q^n}{\mathrm{Vol}_q(d-1)}=\frac{q^n}{\sum_{k=0}^{d-1} \binom{n}{k}(q-1)^k}.$$
Using these keywords should get you started, there are many more results.</p>

CC BY-SA 3.0
2018-03-24T11:04:11.733Z
2018-03-24T11:04:11.733Z
kodlu
<p>If you're not looking for asymptotic results, there are extensive tables that are maintained by researchers.</p>

<p>You can find them at <a href="http://www.codetables.de" rel="nofollow noreferrer">www.codetables.de</a>. Go to that webpage, and click "linear codes". They have complete tables of known upper and lower bounds for linear GF(3) (ternary) codes with $n \leq 243$.</p>

CC BY-SA 3.0
2018-03-26T12:07:43.350Z
2018-03-26T12:13:05.293Z
2018-03-26T12:13:05.293Z
Peter Shor 
<p>I'm not sure I can find an explicit formula. Wondering if anyone can come up with lower/upper bounds.</p>

<coding-theory>
Bounds for maximum number of code-words in a ternary error correcting code with length n and distance d?
Mkch
# Bounds for maximum number of code-words in a ternary error correcting code with length n and distance d?

(Asked by: username_0 on 2018-03-23)

I'm not sure I can find an explicit formula. Wondering if anyone can come up with lower/upper bounds.

**Tags:** coding-theory

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## 2 answers

### Answer by username_1 on 2018-03-24. Score: 2 | Accepted answer

The exact answer is unknown in general. One standard upper bound for $q-$ary codes is the Singleton bound, which gives
$$|C|\leq q^{n-d+1},$$ and codes meeting this bound are called MDS.

A lower bound is the Gilbert-Varshamov bound, given by

$$|C|\geq \frac{q^n}{\mathrm{Vol}\_q(d-1)}=\frac{q^n}{\sum\_{k=0}^{d-1} \binom{n}{k}(q-1)^k}.$$
Using these keywords should get you started, there are many more results.


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### Answer by username_2 on 2018-03-26. Score: 2

If you're not looking for asymptotic results, there are extensive tables that are maintained by researchers.

You can find them at [www.codetables.de](http://www.codetables.de). Go to that webpage, and click "linear codes". They have complete tables of known upper and lower bounds for linear GF(3) (ternary) codes with $n \leq 243$.


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