MinT - Best Known (85, 85+85, s)-Nets in Base 8

Best Known (85, 85+85, s)-Nets in Base 8

(85, 85+85, 194)-Net over F₈ — Constructive and digital

Digital (85, 170, 194)-net over F₈, using

net from sequence [i] based on digital (85, 193)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 85 and N(F) ≥ 194, using
  - F₆ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(85, 85+85, 225)-Net in Base 8 — Constructive

(85, 170, 225)-net in base 8, using

t-expansion [i] based on (83, 170, 225)-net in base 8, using
- 2 times m-reduction [i] based on (83, 172, 225)-net in base 8, using
  - base change [i] based on digital (40, 129, 225)-net over F₁₆, using
    - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
        a function field by GarcÃa and Quoos [i]

(85, 85+85, 272)-Net over F₈ — Digital

Digital (85, 170, 272)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+85, 10125)-Net in Base 8 — Upper bound on s

There is no (85, 170, 10126)-net in base 8, because

1 times m-reduction [i] would yield (85, 169, 10126)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 419 244674 001640 858136 384909 838962 955145 886928 679096 798938 426534 307139 398142 800016 700937 712574 016069 012147 637151 173477 025130 981113 201280 450583 647732 707544 > 8¹⁶⁹ [i]