Embedding in a perfect code

For any 1-error-correcting binary code C of length m we will construct a 1-perfect binary code P (C) of length n = 2 − 1 such that fixing the last n − m coordinates by zeroes in P (C) gives C. In particular, any complete or partial Steiner triple system (or any other system that forms a 1-code) can always be embedded in a 1-perfect code of some length (compare with [13]). Since the weight-3 words of a 1-perfect code P with 0 ∈ P form a Steiner triple system, and the weight-4 words of an extended 1-perfect code P with 0 ∈ P form a Steiner quadruple system, we have, as corollaries, the following well-known facts: a patrial Steiner triple (quadruple) system can always be embedded in a Steiner triple (quadruple) system [18] ([7]) (these results, as well as many other embedding theorems for Steiner systems, can be found in [10, 5]). Notation: