MinT - Best Known (82−27, 82, s)-Nets in Base 8

Best Known (82−27, 82, s)-Nets in Base 8

(82−27, 82, 368)-Net over F₈ — Constructive and digital

Digital (55, 82, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 14, 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]
  - a shift-net [i]
2. digital (41, 68, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 34, 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]

(82−27, 82, 520)-Net in Base 8 — Constructive

(55, 82, 520)-net in base 8, using

8² times duplication [i] based on (53, 80, 520)-net in base 8, using
- base change [i] based on digital (33, 60, 520)-net over F₁₆, using
  - trace code for nets [i] based on digital (3, 30, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(82−27, 82, 1076)-Net over F₈ — Digital

Digital (55, 82, 1076)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−27, 82, 342960)-Net in Base 8 — Upper bound on s

There is no (55, 82, 342961)-net in base 8, because

1 times m-reduction [i] would yield (55, 81, 342961)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 14 135303 064862 354075 105978 361421 304174 105646 521947 499972 607090 240242 749904 > 8⁸¹ [i]