New Constant-Weight Codes from Propagation Rules

This paper proposes some simple propagation rules which give rise to new binary constant-weight codes.

💡 Research Summary

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The paper “New Constant‑Weight Codes from Propagation Rules” introduces a set of simple yet powerful propagation techniques that transform existing q‑ary codes into binary constant‑weight codes, thereby yielding new lower bounds on the maximal size (A(n,d,w)) of such codes. The authors begin by recalling standard definitions: the Hamming distance on (Z_q^n), the weight of a vector as its distance from the zero vector, and the Johnson space (J_n^q(w)) consisting of all vectors of weight (w). A binary constant‑weight code is simply a subset of (J_n(w)=J_n^2(w)) with a prescribed minimum distance.

The core contribution consists of two propagation rules. The first rule (Theorem 2.1) observes that for any q‑ary code (C\subseteq Z_q^n) with length (n) and minimum distance (d), the intersection of a coset (u+C) with the binary Johnson space (J_n(w)) forms a binary constant‑weight code of length (n), weight (w), and minimum distance at least (2\lfloor (d+1)/2\rfloor). By selecting the coset that maximizes the size of this intersection, one obtains a code of size (N). The second rule (Theorem 2.2) replaces the maximization by an averaging argument: over all (q^n) cosets, the average intersection size is (\frac{M\binom{n}{w}}{q^n}) where (M=|C|). Consequently, at least one coset yields a binary constant‑weight code of size not smaller than this average, establishing the lower bound
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