# Binary self dual codes, first order Reed Muller codes

Started by February 22, 2007
```Hello.  I've been trying to learn more about special linear codes such as
Reed-Muller codes, binary self-dual and self-orthogonal codes, and
stumbled upon a couple of problems.

For example, is there a binary self-dual (10,5,4) code?  In some articles
I've read, it was 'shown' that it does exist, but my colleagues say it
does not.  So I am rather confused as to who to believe.  Now just for a
binary (10,5) self dual code, what is an example of a generator matrix for
the code?  Finally, are first order Reed Muller codes self-orthogonal
and/or self-dual?

Basically, I'm confused and having trouble of whether certain types of
codes are self-dual/orthogonal or not.

Some help and clarification would be very much appreciated!

```
```On Feb 22, 6:51 am, "jules" <juliec_1...@yahoo.ca> wrote:
> Reed-Muller codes, binary self-dual and self-orthogonal codes, and
> stumbled upon a couple of problems.
>
> For example, is there a binary self-dual (10,5,4) code?  In some articles
> I've read, it was 'shown' that it does exist, but my colleagues say it
> does not.  So I am rather confused as to who to believe.  Now just for a
> binary (10,5) self dual code, what is an example of a generator matrix for
> the code?  Finally, are first order Reed Muller codes self-orthogonal
> and/or self-dual?
>
> Basically, I'm confused and having trouble of whether certain types of
> codes are self-dual/orthogonal or not.
>
> Some help and clarification would be very much appreciated!

Not sure if this will give a direct answer, but if anybody
knows the existence of self-dual binary codes, it's probably
Neil Sloane:

http://www.research.att.com/~njas/doc/pless.ps

The problem is often it is easier to prove that something
exists (here it is!) rather than it does not exist :-P.

Reed-Muller codes are self-orthogonal, but not all of them
are self-dual.  I think only RM((m-1)/2,m) codes are
self-dual, but not 100% sure.

Julius

```