MinT - Best Known (104−31, 104, s)-Nets in Base 8

Best Known (104−31, 104, s)-Nets in Base 8

(104−31, 104, 416)-Net over F₈ — Constructive and digital

Digital (73, 104, 416)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (21, 36, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 18, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
2. digital (37, 68, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 34, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄ (see above)

(104−31, 104, 576)-Net in Base 8 — Constructive

(73, 104, 576)-net in base 8, using

8 times m-reduction [i] based on (73, 112, 576)-net in base 8, using
- trace code for nets [i] based on (17, 56, 288)-net in base 64, using
  - base change [i] based on digital (9, 48, 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]

(104−31, 104, 2340)-Net over F₈ — Digital

Digital (73, 104, 2340)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(104−31, 104, 1458423)-Net in Base 8 — Upper bound on s

There is no (73, 104, 1458424)-net in base 8, because

1 times m-reduction [i] would yield (73, 103, 1458424)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1042 962912 662819 025964 483589 406236 307722 437276 316714 533804 681204 883372 864994 137290 552147 792426 > 8¹⁰³ [i]