41 research outputs found
Minimizing the regularity of maximal regular antichains of 2- and 3-sets
Let be a natural number. We study the problem to find the
smallest such that there is a family of 2-subsets and
3-subsets of with the following properties: (1)
is an antichain, i.e. no member of is a subset of
any other member of , (2) is maximal, i.e. for every
there is an with or , and (3) is -regular, i.e. every point
is contained in exactly members of . We prove lower
bounds on , and we describe constructions for regular maximal antichains
with small regularity.Comment: 7 pages, updated reference
Long Nonbinary Codes Exceeding the Gilbert - Varshamov Bound for any Fixed Distance
Let A(q,n,d) denote the maximum size of a q-ary code of length n and distance
d. We study the minimum asymptotic redundancy \rho(q,n,d)=n-log_q A(q,n,d) as n
grows while q and d are fixed. For any d and q<=d-1, long algebraic codes are
designed that improve on the BCH codes and have the lowest asymptotic
redundancy \rho(q,n,d) <= ((d-3)+1/(d-2)) log_q n known to date. Prior to this
work, codes of fixed distance that asymptotically surpass BCH codes and the
Gilbert-Varshamov bound were designed only for distances 4,5 and 6.Comment: Submitted to IEEE Trans. on Info. Theor
Improved Bounds for Progression-Free Sets in C^nβ
Let G be a finite group, and let rβ(G) represent the size of the largest subset of G without non-trivial three-term progressions. In a recent breakthrough, Croot, Lev and Pach proved that rβ(Cβ^n) β€ (3.611)^n, where C_m denotes the cyclic group of order m. For finite abelian groups Gβ
β^n_(i=1), where mβ,β¦,m_n denote positive integers such that mβ |β¦|m_n, this also yields a bound of the form rβ(G)β©½(0.903)^(rkβ(G))|G|, with rkβ(G) representing the number of indices i β {1,β¦, n} with 4 |m_i. In particular, rβ(Cnβ) β€ (7.222)^n. In this paper, we provide an exponential improvement for this bound, namely rβ(Cnβ) β€ (7.0899)^n