Using elements of the structural theory of matchings and a recently proved conjecture concerning bricks, it is shown that every n-extendable brick (except K4, C6 and the Petersen graph) with p vertices and q edges contains at least q − p + (n − 1)!! perfect matchings. If the girth of such an n-extendable brick is at least five, then this graph has at least q − p + nn−1 perfect matchings. As a c...

The Golomb-Welch conjecture deals with the existence of perfect eerror correcting Lee codes of word length n, PL(n, e) codes. Although there are many papers on the topic, the conjecture is still far from being solved. In this paper we initiate the study of an invariant connected to abelian groups that enables us to reformulate the conjecture, and then to prove the non-existence of linear PL(n, ...

We say that a real X is n-generic relative to a perfect tree T if X is a path through T and for all Σn(T ) sets S, there exists a number k such that either X|k ∈ S or for all σ ∈ T extending X|k we have σ / ∈ S. A real X is n-generic relative to some perfect tree if there exists such a T . We first show that for every number n all but countably many reals are n-generic relative to some perfect ...

THEOREM 1.1 (U. Sattes). Let r > 2 and g E C[O, l]\B$,‘. Then f”EB$’ is a best approximation to g, in L” (such a best approximation necessari/J) exisrs) if and only if there exists a subinterual (a, /?) c IO. 1 I and a positilse integer M > r + 1 for which the following conditions hold (i) f”l,n.ll, is a Perfect spline of degree r with exactly) M ~ r -1 knots arzd I.f”““(s)l = I a. e. on [u,pI....

Archaeologists working on plantation sites have typically ignored the role that violence played in the lives of enslaved African-Americans. By drawing together examples from archaeological publications in which violence has been mentioned, the commentary illustrates not only both the types and scale of violence endured by African-Americans, but also how archaeologists have effectively sanitized...

In this paper from q-ary perfect codes new completely regular q-ary codes are constructed. In particular, two new ternary completely regular codes are obtained from ternary Golay [11, 6, 5] code. The first [11, 5, 6] code with covering radius ρ = 4 coincides with the dual Golay code and its intersection array is (22, 20, 18, 2, 1; 1, 2, 9, 20, 22) . The second [10, 5, 5] code, with covering rad...