The XOR cut-set structure, and combinatorial designs
You can use a linear code with distance $2k$ or so. The parity check matrix of the code has the property that the XOR of any set of at most $2k-1$ columns is non-zero. This means, in particular, that given the XOR of at most $k$ columns, you can determine (not necessarily efficiently) how many columns were XORed (since if you couldn't you would obtain a set of at most $2k-1$ columns which XORs to zero). The cost of this encoding is the number of rows of the parity check matrix. If you choose the parameters correctly and an efficiently decodable code (recall that the parity check matrix of a code is the generator matrix of its dual), then you probably obtain the conclusion stated in the remark.