MinT - Best Known (126, 126+49, s)-Nets in Base 2

Best Known (126, 126+49, s)-Nets in Base 2

(126, 126+49, 135)-Net over F₂ — Constructive and digital

Digital (126, 175, 135)-net over F₂, using

2¹ times duplication [i] based on digital (125, 174, 135)-net over F₂, using
- trace code for nets [i] based on digital (9, 58, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]

(126, 126+49, 188)-Net over F₂ — Digital

Digital (126, 175, 188)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+49, 1457)-Net in Base 2 — Upper bound on s

There is no (126, 175, 1458)-net in base 2, because

1 times m-reduction [i] would yield (126, 174, 1458)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 24297 663308 575480 830953 203450 578657 690782 244531 139372 > 2¹⁷⁴ [i]