MinT - Best Known (33, 46, s)-Nets in Base 2

Best Known (33, 46, s)-Nets in Base 2

(33, 46, 64)-Net over F₂ — Constructive and digital

Digital (33, 46, 64)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 15, 36)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (18, 31, 32)-net over F₂, using
  - 1 times m-reduction [i] based on digital (18, 32, 32)-net over F₂, using
    - a shift-net [i]

(33, 46, 76)-Net over F₂ — Digital

Digital (33, 46, 76)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁴⁶, 76, F₂, 2, 13) (dual of [(76, 2), 106, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2⁴⁶, 152, F₂, 13) (dual of [152, 106, 14]-code), using
  - adding a parity check bit [i] based on linear OA(2⁴⁵, 151, F₂, 12) (dual of [151, 106, 13]-code), using
    - a â€œMMTâ€ code from Brouwerâ€™s database [i]

(33, 46, 533)-Net in Base 2 — Upper bound on s

There is no (33, 46, 534)-net in base 2, because

1 times m-reduction [i] would yield (33, 45, 534)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 35 382565 800880 > 2⁴⁵ [i]