MinT - Best Known (104, 163, s)-Nets in Base 8

Best Known (104, 163, s)-Nets in Base 8

(104, 163, 371)-Net over F₈ — Constructive and digital

Digital (104, 163, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 31, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (73, 132, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 66, 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]

(104, 163, 576)-Net in Base 8 — Constructive

(104, 163, 576)-net in base 8, using

3 times m-reduction [i] based on (104, 166, 576)-net in base 8, using
- trace code for nets [i] based on (21, 83, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (21, 84, 288)-net in base 64, using
    - base change [i] based on digital (9, 72, 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]

(104, 163, 1131)-Net over F₈ — Digital

Digital (104, 163, 1131)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(104, 163, 184854)-Net in Base 8 — Upper bound on s

There is no (104, 163, 184855)-net in base 8, because

1 times m-reduction [i] would yield (104, 162, 184855)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 199 822877 317141 209267 901869 185211 650623 335352 025401 348700 364726 064509 969268 753021 330906 960060 608711 021277 897524 148410 817063 656915 277252 993008 876416 > 8¹⁶² [i]