Simplifications to "A new approach to the covering radius..."

We simplify the proofs of four results in [3], restating two of them for greater clarity. The main purpose of this note is to give a brief transparent proof of Theorem 7 of [3], the main upper bound of that paper. The secondary purpose is to give a more direct statement and proof of the integer programming determination of covering radius of [3]. Theorem 7 of [3] follows from a simple result in [2], which we state with the notation (for the linear code A) (1) g(A): =a generator matrix of A, t(A) : =the covering radius of A. THEOREM 1 [2]. If A is a code with generator matrix * g(A) = g(Ao) 0 X X then t(A) < t(Ao) + t(A1). To describe the codes A0 and A1 : Pick any subset X of coordinate-places of A. A1 is the projection of A on X; we get Ao from the subcode D of A which vanishes on X by projecting Don X. (Ao [A1] is sometimes called a shortened [punctured] code of A.) Before stating Theorem 2, let us agree that all codes B, C are binary, linear, and have no coordinates identically 0. (The last need not be true of C0 .) We also need the following notation: 1