MinT - Best Known (129−45, 129, s)-Nets in Base 8

Best Known (129−45, 129, s)-Nets in Base 8

(129−45, 129, 378)-Net over F₈ — Constructive and digital

Digital (84, 129, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 25, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (59, 104, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 52, 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]

(129−45, 129, 576)-Net in Base 8 — Constructive

(84, 129, 576)-net in base 8, using

1 times m-reduction [i] based on (84, 130, 576)-net in base 8, using
- trace code for nets [i] based on (19, 65, 288)-net in base 64, using
  - 5 times m-reduction [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]

(129−45, 129, 1117)-Net over F₈ — Digital

Digital (84, 129, 1117)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(129−45, 129, 232311)-Net in Base 8 — Upper bound on s

There is no (84, 129, 232312)-net in base 8, because

1 times m-reduction [i] would yield (84, 128, 232312)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 39 402839 848397 093739 186418 168054 145745 501679 455113 396004 840020 930460 595337 147502 999340 467748 627516 915836 763580 500149 > 8¹²⁸ [i]