MinT - Best Known (44−31, 44, s)-Nets in Base 128

Best Known (44−31, 44, s)-Nets in Base 128

(44−31, 44, 288)-Net over F₁₂₈ — Constructive and digital

Digital (13, 44, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 44, 288)-net over F₁₂₈, using
- net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
    - a function field by SÃ©mirat [i]

(44−31, 44, 321)-Net over F₁₂₈ — Digital

Digital (13, 44, 321)-net over F₁₂₈, using

t-expansion [i] based on digital (12, 44, 321)-net over F₁₂₈, using
- net from sequence [i] based on digital (12, 320)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 12 and N(F) ≥ 321, using
    - a curve from the manYPoints database [i]

(44−31, 44, 55536)-Net in Base 128 — Upper bound on s

There is no (13, 44, 55537)-net in base 128, because

1 times m-reduction [i] would yield (13, 43, 55537)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 4 074997 230273 231491 770389 957157 021075 481956 927165 210829 061135 372450 284898 559046 597807 659216 > 128⁴³ [i]