MinT - Best Known (88−28, 88, s)-Nets in Base 8

Best Known (88−28, 88, s)-Nets in Base 8

(88−28, 88, 379)-Net over F₈ — Constructive and digital

Digital (60, 88, 379)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 18, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (42, 70, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 35, 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]

(88−28, 88, 576)-Net in Base 8 — Constructive

(60, 88, 576)-net in base 8, using

trace code for nets [i] based on (16, 44, 288)-net in base 64, using
- 5 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
  - base change [i] based on digital (9, 42, 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]

(88−28, 88, 1383)-Net over F₈ — Digital

Digital (60, 88, 1383)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(88−28, 88, 410125)-Net in Base 8 — Upper bound on s

There is no (60, 88, 410126)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 29 643702 844242 384476 291149 926878 938011 517437 270588 918822 483090 066892 747299 889104 > 8⁸⁸ [i]