Castro, Francis; Janwa, Heeralal; Mullen, Gary; Rubio, Ivelisse \(e\)-perfect codes. (English) Zbl 1403.94123 Bull. Inst. Comb. Appl. 75, 83-90 (2015). From the text: Perfect codes provide one of the most important and widely studied classes of error-correcting codes. the problem of classifying the parameters of perfect codes remained open until 1973. In this paper we generalize perfect codes to \(e\)-perfect codes over finite fields. We present a list of parameters for \(e\)-perfect codes and conjecture that the list is complete, provide constructions for almost all the \(e\)-perfect codes listed in the conjecture and present partial results on proving that there are no other parameters for \(e\)-perfect codes.Definition: Let \(0\leq e\leq n\) be an integer. A \(t\)-error correcting code \(C\) with parameters \((n,M,d)\), \(t=\lfloor \frac {d-1}{2}\rfloor\), is said to be \(e\)-perfect if in the Hamming bound equality is achieved when, on the right-hand side, \(q^n\) is replaced by \(q^e\). Thus an \(n\)-perfect code is a perfect code. MSC: 94B25 Combinatorial codes 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) Keywords:generalization of perfect codes; \(e\)-perfect codes; list of parameters; constructions for almost all the \(e\)-perfect codes PDFBibTeX XMLCite \textit{F. Castro} et al., Bull. Inst. Comb. Appl. 75, 83--90 (2015; Zbl 1403.94123)