MinT - Best Known (57, 57+23, s)-Nets in Base 8

Best Known (57, 57+23, s)-Nets in Base 8

(57, 57+23, 416)-Net over F₈ — Constructive and digital

Digital (57, 80, 416)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (17, 28, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 14, 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 (29, 52, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 26, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄ (see above)

(57, 57+23, 576)-Net in Base 8 — Constructive

(57, 80, 576)-net in base 8, using

4 times m-reduction [i] based on (57, 84, 576)-net in base 8, using
- trace code for nets [i] based on (15, 42, 288)-net in base 64, using
  - base change [i] based on digital (9, 36, 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]

(57, 57+23, 2498)-Net over F₈ — Digital

Digital (57, 80, 2498)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+23, 2146559)-Net in Base 8 — Upper bound on s

There is no (57, 80, 2146560)-net in base 8, because

1 times m-reduction [i] would yield (57, 79, 2146560)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 220856 762589 332307 066403 276370 158795 782098 235657 747531 015253 853313 675041 > 8⁷⁹ [i]