Macwilliams Identities and Coordinate Partitions

MacWilliams Identities and Matroid Polynomials

“…The following theorem is the MacWilliams identity for support enumerators. A proof will be provided in Section 4, but for now remark that it follows from an equivalent result, Proposition 2 in [14], or from stronger results such as Theorem 7 below or Theorem 14 in [9, Ch. 5.…”

No Projective 16-Divisible Binary Linear Code of Length 131 Exists

“…The weight enumerator of a linear [n, k] code C can be refined to a so-called partition weight enumerator, see e.g. [7]. To this end let r ≥ 1 be an integer and ∪ r j=1 P j be a partition of the coordinates {1, .…”

Optimal doubly constant weight codes

“…This partition of the coordinates has appeared in [16] and also handled in [33] is the one required for description of doubly constant weight codes. For the rest of this article, we always assume that for a (w 1 , n 1 , w 2 , n 2 , d) code, the first set of n 1 coordinates is A and the second set of n 2 coordinates is B.…”