MinT - Best Known (84, 169, s)-Nets in Base 8

Best Known (84, 169, s)-Nets in Base 8

(84, 169, 130)-Net over F₈ — Constructive and digital

Digital (84, 169, 130)-net over F₈, using

t-expansion [i] based on digital (76, 169, 130)-net over F₈, using
- 3 times m-reduction [i] based on digital (76, 172, 130)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (14, 62, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
    2. digital (14, 110, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(84, 169, 225)-Net in Base 8 — Constructive

(84, 169, 225)-net in base 8, using

t-expansion [i] based on (83, 169, 225)-net in base 8, using
- 3 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]

(84, 169, 264)-Net over F₈ — Digital

Digital (84, 169, 264)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(84, 169, 9635)-Net in Base 8 — Upper bound on s

There is no (84, 169, 9636)-net in base 8, because

1 times m-reduction [i] would yield (84, 168, 9636)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 52 499720 345098 685181 759458 879038 643346 787673 228451 603563 499090 557771 296575 189399 942213 169670 529628 652601 766619 657731 525114 011686 992854 336406 166789 051312 > 8¹⁶⁸ [i]