MinT - Best Known (84, 84+89, s)-Nets in Base 8

Best Known (84, 84+89, s)-Nets in Base 8

(84, 84+89, 130)-Net over F₈ — Constructive and digital

Digital (84, 173, 130)-net over F₈, using

t-expansion [i] based on digital (76, 173, 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, 111, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(84, 84+89, 225)-Net in Base 8 — Constructive

(84, 173, 225)-net in base 8, using

8¹ times duplication [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, 84+89, 246)-Net over F₈ — Digital

Digital (84, 173, 246)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(84, 84+89, 8329)-Net in Base 8 — Upper bound on s

There is no (84, 173, 8330)-net in base 8, because

1 times m-reduction [i] would yield (84, 172, 8330)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 214617 141813 480910 289369 216245 130657 514434 303688 430914 343618 748433 143411 465927 460336 924326 949412 719336 152052 689830 136429 793592 178971 686515 991936 575081 777568 > 8¹⁷² [i]