MinT - Best Known (109−31, 109, s)-Nets in Base 5

Best Known (109−31, 109, s)-Nets in Base 5

(109−31, 109, 296)-Net over F₅ — Constructive and digital

Digital (78, 109, 296)-net over F₅, using

9 times m-reduction [i] based on digital (78, 118, 296)-net over F₅, using
- trace code for nets [i] based on digital (19, 59, 148)-net over F₂₅, using
  - net from sequence [i] based on digital (19, 147)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 19 and N(F) ≥ 148, using
      - a function field by GarcÃa and Quoos [i]

(109−31, 109, 1058)-Net over F₅ — Digital

Digital (78, 109, 1058)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−31, 109, 173086)-Net in Base 5 — Upper bound on s

There is no (78, 109, 173087)-net in base 5, because

1 times m-reduction [i] would yield (78, 108, 173087)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3081 566832 541125 235744 284034 715482 847263 220546 135477 222721 690451 540600 767045 > 5¹⁰⁸ [i]