MinT - Best Known (89−30, 89, s)-Nets in Base 8

Best Known (89−30, 89, s)-Nets in Base 8

(89−30, 89, 363)-Net over F₈ — Constructive and digital

Digital (59, 89, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (44, 74, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 37, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(89−30, 89, 520)-Net in Base 8 — Constructive

(59, 89, 520)-net in base 8, using

8¹ times duplication [i] based on (58, 88, 520)-net in base 8, using
- base change [i] based on digital (36, 66, 520)-net over F₁₆, using
  - trace code for nets [i] based on digital (3, 33, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(89−30, 89, 1000)-Net over F₈ — Digital

Digital (59, 89, 1000)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−30, 89, 209403)-Net in Base 8 — Upper bound on s

There is no (59, 89, 209404)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 237 153755 114940 834902 841328 284846 812914 116017 106943 635066 786835 724508 453646 932772 > 8⁸⁹ [i]