# Is there a decimal perfect code encoding 9 digits with 11 digits with a minimum distance of 3

Since $\binom{11}{0} 9^0 + \binom{11}{1} 9^1 = 100$ there might be a perfect decimal code encoding 9 digits with 11 digits with minimum distance of 3. Is there?

• There is no finite field with $10$ elements and so $\mathbb F_{10}$ is an abuse of notation when used to denote the set you are talking about.. Commented Feb 25, 2017 at 20:04
• I never said such a set was - or had to be - a vector space. Commented Feb 27, 2017 at 6:49
• My comment was in response to two other (now-deleted) comments which used $\mathbb F_{10}$. If I recall correctly, the first comment (by someone else) used the notation $\mathbb F_{10}$ and your response did not push back on it. Both comments have now gone forever. I will aver, though, that in coding theory circles, $[n, k, d]_q$ is generally considered as denoting a linear code of length $n$, dimension $k$ and minimum distance $d$ over the finite field $\mathbb F_q$, and so in a sense, you have suggested that you think the set under consideration is a vector space. Commented Feb 27, 2017 at 13:09
• Thanks, I didn't know that the notation implied a finite field, or even a linear code, I'll update the question. Commented Feb 27, 2017 at 17:15

A survey if perfect codes (J H Van Lint - 1975) states as Problem 2.7 (p. 205) :

Are there any perfect single-error-correcting codes over an alphabet $F$ for which $|F|$ is not a power of a prime?

(recall that "single-error-correcting" is equivalent to "minimum distance of 3")

According to the paper, there is no example of such a code known. And it cites the smaller candidate example $q=6$, $n=7$ to say that that is the this is the only case for which it has been shown that there is no perfect s.e.c. code.

Perhaps there are more conclusive results in more recent literature.

• This is surprising, other examples could also be shown not to have perfect codes by enumeration. Simple, if time consuming. Are there non trivial examples of perfect error correcting codes (correcting one or more error) over non prime power alphabets? Commented Mar 3, 2017 at 1:23
• (the link seems busted) Commented Mar 6, 2017 at 17:48
• @ArthurB. Sorry. Fixed. Commented Mar 6, 2017 at 19:40