MinT - Best Known (129−46, 129, s)-Nets in Base 8

Best Known (129−46, 129, s)-Nets in Base 8

(129−46, 129, 363)-Net over F₈ — Constructive and digital

Digital (83, 129, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 23, 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 (60, 106, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 53, 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]

(129−46, 129, 518)-Net in Base 8 — Constructive

(83, 129, 518)-net in base 8, using

8¹ times duplication [i] based on (82, 128, 518)-net in base 8, using
- base change [i] based on digital (50, 96, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 48, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(129−46, 129, 1000)-Net over F₈ — Digital

Digital (83, 129, 1000)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(129−46, 129, 156485)-Net in Base 8 — Upper bound on s

There is no (83, 129, 156486)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 315 216226 223616 065821 515564 267262 934671 639497 957679 323597 394072 658037 401513 546110 085499 709417 421562 275536 303570 116128 > 8¹²⁹ [i]