Perfect error correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of $e-$error correcting perfect codes over $q-$ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect $e-$error correcting codes were found if $e \ge 3$, and it was conjectured that no perfect $2-$error correcting codes exist over any $q-$ary alphabet, where $q > 3$. In the 1970s, this was proved for $q$ a prime power, for $q = 2^r3^s$ and for only $7$ other values of $q$. Almost $50$ years later, it is surprising to note that there have been no new results in this regard and the classification of $2-$error correcting codes over non-prime power alphabets remains an open problem. In this paper, we use techniques from the resolution of generalised Ramanujan--Nagell equation and from modern computational number theory to show that perfect $2-$error correcting codes do not exist for $172$ new values of $q$ which are not prime powers, substantially increasing the values of $q$ which are now classified. In addition, we prove that, for any fixed value of $q$, there can be at most finitely many perfect $2-$error correcting codes over an alphabet of size $q$.

翻译：完美纠错码能够在保证纠错能力的同时实现信息的最优传输。因此，证明其存在性一直是纯数学和信息论领域的经典问题。事实上，在二十世纪末，对$q$元字母表上$e-$纠错完美码参数的分类曾是一个极为活跃的研究课题。最终，当$e \ge 3$时所有完美$e-$纠错码的参数均已找到，且学界猜想在任何$q > 3$的$q$元字母表上均不存在完美$2-$纠错码。在1970年代，该结论对于$q$为素数幂、$q = 2^r3^s$以及其他仅$7$个$q$值得到了证明。近$50$年后，值得注意的是该领域再无新进展，非素数幂字母表上$2-$纠错码的分类问题仍然悬而未决。本文运用广义拉马努金-内格尔方程的求解技术及现代计算数论方法，证明了在$172$个新的非素数幂$q$值上不存在完美$2-$纠错码，从而显著扩展了已完成分类的$q$值范围。此外，我们证明了对于任意固定的$q$值，在大小为$q$的字母表上最多只能存在有限多个完美$2-$纠错码。