MinT - Best Known (72−32, 72, s)-Nets in Base 8

Best Known (72−32, 72, s)-Nets in Base 8

(72−32, 72, 208)-Net over F₈ — Constructive and digital

Digital (40, 72, 208)-net over F₈, using

2 times m-reduction [i] based on digital (40, 74, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 37, 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]

(72−32, 72, 258)-Net over F₈ — Digital

Digital (40, 72, 258)-net over F₈, using

trace code for nets [i] based on digital (4, 36, 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]

(72−32, 72, 11245)-Net in Base 8 — Upper bound on s

There is no (40, 72, 11246)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 105429 088897 351657 301291 195991 470918 445213 314080 847116 621126 055958 > 8⁷² [i]