MinT - Best Known (56, 56+48, s)-Nets in Base 8

Best Known (56, 56+48, s)-Nets in Base 8

(56, 56+48, 208)-Net over F₈ — Constructive and digital

Digital (56, 104, 208)-net over F₈, using

2 times m-reduction [i] based on digital (56, 106, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 53, 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]

(56, 56+48, 258)-Net over F₈ — Digital

Digital (56, 104, 258)-net over F₈, using

trace code for nets [i] based on digital (4, 52, 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]

(56, 56+48, 11457)-Net in Base 8 — Upper bound on s

There is no (56, 104, 11458)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8352 659579 116215 292011 669889 564548 695869 758661 800058 685615 724607 588781 416550 478338 888826 169688 > 8¹⁰⁴ [i]