MinT - Best Known (53, 53+45, s)-Nets in Base 8

Best Known (53, 53+45, s)-Nets in Base 8

(53, 53+45, 208)-Net over F₈ — Constructive and digital

Digital (53, 98, 208)-net over F₈, using

2 times m-reduction [i] based on digital (53, 100, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 50, 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]

(53, 53+45, 258)-Net over F₈ — Digital

Digital (53, 98, 258)-net over F₈, using

trace code for nets [i] based on digital (4, 49, 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]

(53, 53+45, 12390)-Net in Base 8 — Upper bound on s

There is no (53, 98, 12391)-net in base 8, because

1 times m-reduction [i] would yield (53, 97, 12391)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3983 277176 479632 165113 885200 615622 085188 635923 533637 591571 928992 091590 987181 815459 590602 > 8⁹⁷ [i]