In this paper, the problem of correction of a single error in $q$-ary symmetric channel with noiseless feedback is considered. We propose an algorithm to construct codes with feedback inductively. For all prime power $q$ we prove that two instances of feedback are sufficient to transmit over the $q$-ary symmetric channel the same number of messages as in the case of complete feedback. Our other contribution is the construction of codes with one-time feedback with the same parameters as Hamming codes for $q$ that is not a prime power. We also construct single-error-correcting codes with one-time feedback of size $q^{n-2}$ for arbitrary $q$ and $n\leq q+1$, which can be seen as an analog for Reed-Solomon codes.

翻译：本文研究了无噪声反馈条件下$q$元对称信道中单个错误的纠正问题。我们提出了一种通过归纳法构造带反馈编码的算法。对于所有素数幂$q$，我们证明：在$q$元对称信道上传输与完全反馈情况下相同数量的信息时，仅需两次反馈实例即可实现。另一项贡献是：针对非素数幂的$q$，我们构造了参数与汉明码相同的单次反馈编码。此外，对于任意$q$及$n\leq q+1$，我们构造了大小为$q^{n-2}$的单次反馈单错纠正码，该编码可视为里德-所罗门码的类比方案。