A Scheme to Construct Distance Three Codes Using Latin Squares, with Applications to the n-Cube

Let B,, denote the set of n-bit binary strings, and let Q, denote the graph of the n-cube where V(Q,> = B, and where two vertices are adjacent iff their Humming distance is exactly one. A subset C of B, is called a code, and the elements of C are referred to as codewords. C is said to be a linear code if the codeword obtained from component-wise sum (modulo 2) of any two elements of C is again in C; otherwise it is a nonlinear code. By a distance-three code is meant a code in which the Hamming distance between any two distinct codewords is at least three. Distance-three codes possess the capability to correct one error and detect two or fewer errors. It is known that if n is of the form 2k 1, then B, admits of a partition into equal-size sets V O,. . . , V, such that each F is a distance-three code and is maximal with respect to this property