MinT - Best Known (89, 89+69, s)-Nets in Base 2

Best Known (89, 89+69, s)-Nets in Base 2

(89, 89+69, 54)-Net over F₂ — Constructive and digital

Digital (89, 158, 54)-net over F₂, using

trace code for nets [i] based on digital (10, 79, 27)-net over F₄, using
- net from sequence [i] based on digital (10, 26)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
    - an explicitly constructive algebraic function field [i]

(89, 89+69, 62)-Net over F₂ — Digital

Digital (89, 158, 62)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+69, 285)-Net in Base 2 — Upper bound on s

There is no (89, 158, 286)-net in base 2, because

1 times m-reduction [i] would yield (89, 157, 286)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 201796 888997 609740 831108 236469 099442 207056 108272 > 2¹⁵⁷ [i]