MinT - Best Known (43−13, 43, s)-Nets in Base 2

Best Known (43−13, 43, s)-Nets in Base 2

(43−13, 43, 55)-Net over F₂ — Constructive and digital

Digital (30, 43, 55)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 14, 26)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (16, 29, 29)-net over F₂, using
  - a shift-net [i]

(43−13, 43, 65)-Net over F₂ — Digital

Digital (30, 43, 65)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁴³, 65, F₂, 2, 13) (dual of [(65, 2), 87, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2⁴³, 130, F₂, 13) (dual of [130, 87, 14]-code), using
  - adding a parity check bit [i] based on linear OA(2⁴², 129, F₂, 12) (dual of [129, 87, 13]-code), using
    - a â€œMMTâ€ code from Brouwerâ€™s database [i]

(43−13, 43, 374)-Net in Base 2 — Upper bound on s

There is no (30, 43, 375)-net in base 2, because

1 times m-reduction [i] would yield (30, 42, 375)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 4 412732 846201 > 2⁴² [i]