MinT - Best Known (128−81, 128, s)-Nets in Base 5

Best Known (128−81, 128, s)-Nets in Base 5

(128−81, 128, 78)-Net over F₅ — Constructive and digital

Digital (47, 128, 78)-net over F₅, using

t-expansion [i] based on digital (38, 128, 78)-net over F₅, using
- net from sequence [i] based on digital (38, 77)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 38 and N(F) ≥ 78, using
    - an explicitly constructive algebraic function field [i]

(128−81, 128, 92)-Net over F₅ — Digital

Digital (47, 128, 92)-net over F₅, using

net from sequence [i] based on digital (47, 91)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 47 and N(F) ≥ 92, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(128−81, 128, 624)-Net in Base 5 — Upper bound on s

There is no (47, 128, 625)-net in base 5, because

1 times m-reduction [i] would yield (47, 127, 625)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 62347 134558 737535 016414 279566 277334 025680 626726 984661 210515 487321 952878 360800 455237 173601 > 5¹²⁷ [i]