MinT - Best Known (86−28, 86, s)-Nets in Base 8

Best Known (86−28, 86, s)-Nets in Base 8

(86−28, 86, 371)-Net over F₈ — Constructive and digital

Digital (58, 86, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 16, 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 (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]

(86−28, 86, 522)-Net in Base 8 — Constructive

(58, 86, 522)-net in base 8, using

trace code for nets [i] based on (15, 43, 261)-net in base 64, using
- 1 times m-reduction [i] based on (15, 44, 261)-net in base 64, using
  - base change [i] based on digital (4, 33, 261)-net over F₂₅₆, using
    - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(86−28, 86, 1188)-Net over F₈ — Digital

Digital (58, 86, 1188)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−28, 86, 304719)-Net in Base 8 — Upper bound on s

There is no (58, 86, 304720)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 463177 272673 924192 420815 771613 714803 146110 396860 763252 311975 839888 896724 537287 > 8⁸⁶ [i]