How to find the minimum string length to produce a set of a given size with a minimum pairwise Hamming distance - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:08:42Z http://mathoverflow.net/feeds/question/104309 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104309/how-to-find-the-minimum-string-length-to-produce-a-set-of-a-given-size-with-a-min How to find the minimum string length to produce a set of a given size with a minimum pairwise Hamming distance unknown (google) 2012-08-08T21:33:10Z 2012-08-08T23:34:58Z <p>Given an alphabet of $q \ge 2$ letters, I want to construct a set $S$ of $x$ strings (of uniform length) such that the minimum Hamming distance between any two strings is $d$. What I need to figure out is the minimum string length $n$ that could produce such a set.</p> <p>When $d = 1$, then $n = ceiling(log_q(x))$, but I can't figure out how to find $n$ for an arbitrary value of $d > 1$.</p> http://mathoverflow.net/questions/104309/how-to-find-the-minimum-string-length-to-produce-a-set-of-a-given-size-with-a-min/104313#104313 Answer by tergi for How to find the minimum string length to produce a set of a given size with a minimum pairwise Hamming distance tergi 2012-08-08T22:41:53Z 2012-08-08T23:05:12Z <p>According to my interpretation of Table I of <a href="http://neilsloane.com/doc/Me54.pdf" rel="nofollow">http://neilsloane.com/doc/Me54.pdf</a> it was not known at the time whether you needed bit ($q=2$) strings of length $n=23$, $n=22$, or possibly only of length $n=21$, to construct a set of $x=50$ codewords that are separated from each other by at least Hamming distance $d=10$. This particular example may or may not still be an open question, but there is probably not a known general formula for $n$ in terms of $q$, $x$, and $d$.</p> <p>Edit: The bounds update at <a href="http://webfiles.portal.chalmers.se/s2/research/kit/bounds/unr.html" rel="nofollow">http://webfiles.portal.chalmers.se/s2/research/kit/bounds/unr.html</a> shows that $n$ is now known to be $22$ for the example above, but you can still see that a nice way to compute the function you want has not been discovered.</p> http://mathoverflow.net/questions/104309/how-to-find-the-minimum-string-length-to-produce-a-set-of-a-given-size-with-a-min/104314#104314 Answer by quid for How to find the minimum string length to produce a set of a given size with a minimum pairwise Hamming distance quid 2012-08-08T23:29:21Z 2012-08-08T23:34:58Z <p>For $q$ the size of the alphabet and $n$ the length of the code it is costumary to denote by <code>$A_q(n,d)$</code> the maximal size of a code with minimum distance $d$. </p> <p>There are numerous investigations on this. tergi already mentioned tables of explicit values. There are however also general bounds known. In particular a classical result is the Gilbert-Varshamov bound that says <code>$$A_q (n,d) \ge \frac{q^n}{\sum_{j=0}^{d-1} C(n,j) (q-1)^j}$$</code> where by $C(n,j)$ I just mean the binomial coefficient but momentarily fail to typeset it properly.</p> <p>This is not precisely what you need as you have some $x$ given that corresponds to the <code>$A_q(n,d)$</code> and need to find a suitable $n$. But for concrete values it would now be easy to solve your problem, and if you need explicit bounds they would (with some additional loss) also be obtainable.</p> <p>Another question would be how to effectively construct the set then. (Depending on what you are trying to achieve there might be different things to consider.) Yet, without further details from you it is hard/impossible to know what type of information would be most useful. </p>