4
$\begingroup$

My professor told me that dimension of bch code is unknown in general, so I made a loop in SAGEMATH that create a BCH code of length $p^m-1$ over $GF(p)$ with every possible designed minimum distance $\delta$, and I asked for it dimension.

The output file ALWAYS ends with a $p^{m+1}$ code of dimension $1$ then there are no code of dimension less then $p+1$ so there are a "jump" of $p$. in fact the "jump" is ALWAYS a divisor of $m$ and there are a certain number of those. and things get nicer when $m$ is prime.

There is well behaved pattern here but I'm not too smart to tell a general formula. But I believe it can be done.

So my question is: Why the dimension of bch code still unknown?

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ the general formulas which exist for the dimension all cease to apply when $\delta$ becomes larger than some $m$ and $p$ dependent bound; you can find several of these formulas here, including a recursion relation for the dimension when $\delta$ is a power of $p$. $\endgroup$
    – Carlo Beenakker
    Jul 1, 2018 at 19:04

1 Answer 1

Reset to default
5
$\begingroup$

This is a problem that coding theorists have been considering since the late 1950's, both for BCH, and more generally for cyclic codes.

The problem is combinatorial in nature, tightly bound to polynomial factorisation over finite fields, and algebraic techniques only go so far.

There are the BCH, Roos, Hartmann-Tzeng bounds, the van Lint-Wilson shift method, but no general closed form formula exists.

There is a nice survey by Ruud Pelikaan here.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.