Are there explicit constructions of codes attaining the GV bound?

Question

I am looking for a code with the following parameters:

• Has $k= 128$ and a rate $R$ of at least $1/20$
• Has a relative distance of at least $\delta = 3/10$
• Provides efficient encoding
• Is binary

This basically boils down to codes that attain the GV bound. I ruled out random linear codes, because I need to be sure that the code really has the required minimal distance.

I am relatively new to coding theory and the best approach I could find so far were algebraic geometric codes, because I read in some places that they attain or even beat the GV bound. However, in some of the theorems I also read that the proofs were not constructive, but only probabilistic.

Hence, before diving into algebraic geomtry, I have the following questions:

1. Are there efficient constructions of algebraic geometric codes attaining the GV bound with the parameters listed above.

2. Are there maybe other codes that achieve the parameters above? I also saw the paper Explicit, almost optimal, epsilon-balanced codes. It has a lot of asymptotics in it, but since my $n$ and $k$ are relatively large compared to usual applications, I thought it might work out.

Answer 1

Edit: even though I still think going out to the GV bound and AG codes may be overkill, I did mistakenly apply this result. Keeping this answer up so no one else makes the same mistake.

Short Answer:

1. Your code search does not actually boil down to looking at the GV bound.
2. Using a bunch of binary Golay codes together could work well enough.

Long Answer:

If we look at a picture of some bounds in coding theory, along with your search criterion for $\delta$ and $R$, we see that there are a whole bunch of codes that might satisfy your conditions (above and to the right of the blue lines in the picture).

Because of this, we may not need to delve into AG codes, which would certainly not be friendly to someone starting out in coding theory, and in general are not defined over $\mathbb{F}_2$ directly. However, there are other classes of codes that are defined over $\mathbb{F}_2$; in particular, let's look at the binary Golay code.

The binary Golay code is a $[24,12,8]_2$ code. As such it has rate $R = 0.5$ and relative minimum distance $\delta = \frac{1}{3} \approx 0.33$. Now we meet three of the four conditions laid out; the last to satisfy is dimension $k=128$.

There are two ways to think of achieving this:

• Separate your $128$ message bits into strings of length $12$ (padding with an additional $4$ bits), and encode each block with the binary Golay code.
• Concatenate the binary Golay code with itself $11$ times, and encode a string containing your $128$ message bits followed again by $4$ padding bits. This big concatenated code retains the same rate, as well as the same relative minimum distance, which can be seen through a small proof.

Of course this doesn't exactly answer your question, since this code has $k=132$, but if you're unhappy with padding $4$ additional bits, I'm sure some kind of modification on the last concatenated piece might make it work out.