MinT - Best Known (102−35, 102, s)-Nets in Base 8

Best Known (102−35, 102, s)-Nets in Base 8

(102−35, 102, 368)-Net over F₈ — Constructive and digital

Digital (67, 102, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 18, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (49, 84, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 42, 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]

(102−35, 102, 520)-Net in Base 8 — Constructive

(67, 102, 520)-net in base 8, using

trace code for nets [i] based on (16, 51, 260)-net in base 64, using
- 1 times m-reduction [i] based on (16, 52, 260)-net in base 64, using
  - base change [i] based on digital (3, 39, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(102−35, 102, 1007)-Net over F₈ — Digital

Digital (67, 102, 1007)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(102−35, 102, 237818)-Net in Base 8 — Upper bound on s

There is no (67, 102, 237819)-net in base 8, because

1 times m-reduction [i] would yield (67, 101, 237819)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 16 297208 861771 650460 326439 382330 716881 022237 532091 772937 411005 451037 692825 353700 175590 918710 > 8¹⁰¹ [i]