On the reconstruction of perfect codes (Q1849959)

For a perfect 1-error correcting binary code \(C\), let \(C(k)\) denote the set of words of \(C\) of weight \(k\). \textit{S. V. Avgustinovich} [Discrete Anal. Issled. Oper. 2, 4-6 (1995; Zbl 0846.94017, Zbl 0861.94018)] has shown that if \(C((n+ 1)/2)= C'((n+ 1)/2)\) for two binary perfect 1-error correcting codes \(C\) and \(C'\) of length \(n\) then \(C= C'\). His proof did not show how to reconstruct a perfect 1-error correcting binary code of length \(n\) from the words of weight \((n+1)/2\). In the present paper this problem is solved.