MinT - Best Known (124−75, 124, s)-Nets in Base 5

Best Known (124−75, 124, s)-Nets in Base 5

(124−75, 124, 82)-Net over F₅ — Constructive and digital

Digital (49, 124, 82)-net over F₅, using

t-expansion [i] based on digital (48, 124, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(124−75, 124, 96)-Net over F₅ — Digital

Digital (49, 124, 96)-net over F₅, using

net from sequence [i] based on digital (49, 95)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 49 and N(F) ≥ 96, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(124−75, 124, 744)-Net in Base 5 — Upper bound on s

There is no (49, 124, 745)-net in base 5, because

1 times m-reduction [i] would yield (49, 123, 745)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 94 951230 370668 838753 498458 591568 309361 500141 511698 788457 370365 628511 161631 038593 012053 > 5¹²³ [i]