MinT - Best Known (128−38, 128, s)-Nets in Base 8

Best Known (128−38, 128, s)-Nets in Base 8

(128−38, 128, 484)-Net over F₈ — Constructive and digital

Digital (90, 128, 484)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (19, 38, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 19, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. digital (52, 90, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 45, 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]

(128−38, 128, 576)-Net in Base 8 — Constructive

(90, 128, 576)-net in base 8, using

t-expansion [i] based on (89, 128, 576)-net in base 8, using
- 12 times m-reduction [i] based on (89, 140, 576)-net in base 8, using
  - trace code for nets [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(128−38, 128, 2805)-Net over F₈ — Digital

Digital (90, 128, 2805)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−38, 128, 1374321)-Net in Base 8 — Upper bound on s

There is no (90, 128, 1374322)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 39 402152 512879 739090 897303 363191 251004 826919 448871 151988 227368 468539 096121 983919 545856 009054 189081 232548 506896 380912 > 8¹²⁸ [i]