MinT - Best Known (85, 85+59, s)-Nets in Base 5

Best Known (85, 85+59, s)-Nets in Base 5

(85, 85+59, 252)-Net over F₅ — Constructive and digital

Digital (85, 144, 252)-net over F₅, using

6 times m-reduction [i] based on digital (85, 150, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 75, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
      - the Hermitian function field over F₂₅ [i]

(85, 85+59, 284)-Net over F₅ — Digital

Digital (85, 144, 284)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+59, 8139)-Net in Base 5 — Upper bound on s

There is no (85, 144, 8140)-net in base 5, because

1 times m-reduction [i] would yield (85, 143, 8140)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 8997 339860 283641 245495 309433 744572 866393 020752 300643 652460 945731 559658 205026 721500 949393 846467 509425 > 5¹⁴³ [i]