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\(e\)-perfect codes. (English) Zbl 1403.94123

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)
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