MinT - Best Known (28, 39, s)-Nets in Base 2

Best Known (28, 39, s)-Nets in Base 2

(28, 39, 58)-Net over F₂ — Constructive and digital

Digital (28, 39, 58)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 12, 36)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (16, 27, 29)-net over F₂, using
  - 2 times m-reduction [i] based on digital (16, 29, 29)-net over F₂, using
    - a shift-net [i]

(28, 39, 80)-Net over F₂ — Digital

Digital (28, 39, 80)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2³⁹, 80, F₂, 2, 11) (dual of [(80, 2), 121, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2³⁹, 160, F₂, 11) (dual of [160, 121, 12]-code), using
  - a â€œGraâ€ code from Grasslâ€™s database [i]

(28, 39, 498)-Net in Base 2 — Upper bound on s

There is no (28, 39, 499)-net in base 2, because

1 times m-reduction [i] would yield (28, 38, 499)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 276308 478096 > 2³⁸ [i]