MinT - Best Known (31−18, 31, s)-Nets in Base 2

Best Known (31−18, 31, s)-Nets in Base 2

(31−18, 31, 15)-Net over F₂ — Constructive and digital

Digital (13, 31, 15)-net over F₂, using

net from sequence [i] based on digital (13, 14)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 13 and N(F) ≥ 15, using
  - an explicitly constructive algebraic function field [i]
Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]

(31−18, 31, 32)-Net over F₂ — Upper bound on s (digital)

There is no digital (13, 31, 33)-net over F₂, because

2 times m-reduction [i] would yield digital (13, 29, 33)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁹, 33, F₂, 16) (dual of [33, 4, 17]-code), but
  - Griesmer bound [i]

(31−18, 31, 33)-Net in Base 2 — Upper bound on s

There is no (13, 31, 34)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2323 242356 > 2³¹ [i]