<p>I am exploring explicit constructions of binary linear error-correcting codes and particularly interested in finding explicit linear codes that come closest to achieve:</p>
<ol>
<li>Distance: <span class="math-container">$d = O(log(n))$</span>.</li>
<li>Rate: <span class="math-container">$r = 1 - \frac{O(\log(n))}{n}$</span>.</li>
</ol>
<p>I assume that such a linear code is not known. What is the best rate I can achieve as long as the distance is <span class="math-container">$O(log(n))$</span>?</p>

CC BY-SA 4.0
2024-11-17T07:21:37.293Z
BCH has rate $1 - \frac{O(\log(n)^2)}n$, if I understand correctly, as does the binary Goppa code
Command Master
CC BY-SA 4.0
2024-11-18T04:44:46.923Z
Reed-Muller also gets this rate
Command Master
<coding-theory>
Best-known explicit binary error-correcting codes
OriFrid
# Best-known explicit binary error-correcting codes

(Asked by: username_0 on 2024-11-15)

I am exploring explicit constructions of binary linear error-correcting codes and particularly interested in finding explicit linear codes that come closest to achieve:

1. Distance: $d = O(log(n))$.
2. Rate: $r = 1 - \frac{O(\log(n))}{n}$.

I assume that such a linear code is not known. What is the best rate I can achieve as long as the distance is $O(log(n))$?

**Tags:** coding-theory
> **Comments:**
> **username_1**: BCH has rate $1 - \frac{O(\log(n)^2)}n$, if I understand correctly, as does the binary Goppa code  
> **username_1**: Reed-Muller also gets this rate  

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## 0 answers

