MinT - Best Known (82−37, 82, s)-Nets in Base 8

Best Known (82−37, 82, s)-Nets in Base 8

(82−37, 82, 208)-Net over F₈ — Constructive and digital

Digital (45, 82, 208)-net over F₈, using

2 times m-reduction [i] based on digital (45, 84, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 42, 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]

(82−37, 82, 258)-Net over F₈ — Digital

Digital (45, 82, 258)-net over F₈, using

trace code for nets [i] based on digital (4, 41, 129)-net over F₆₄, using
- net from sequence [i] based on digital (4, 128)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 4 and N(F) ≥ 129, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(82−37, 82, 12489)-Net in Base 8 — Upper bound on s

There is no (45, 82, 12490)-net in base 8, because

1 times m-reduction [i] would yield (45, 81, 12490)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 14 140736 762111 201936 400090 767677 855225 287227 865090 810103 358854 875983 444816 > 8⁸¹ [i]