MinT - Best Known (72, 72+31, s)-Nets in Base 8

Best Known (72, 72+31, s)-Nets in Base 8

(72, 72+31, 402)-Net over F₈ — Constructive and digital

Digital (72, 103, 402)-net over F₈, using

8¹ times duplication [i] based on digital (71, 102, 402)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 26, 48)-net over F₈, using
    - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
        a function field by SÃ©mirat [i]
  2. digital (45, 76, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 38, 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]

(72, 72+31, 576)-Net in Base 8 — Constructive

(72, 103, 576)-net in base 8, using

7 times m-reduction [i] based on (72, 110, 576)-net in base 8, using
- trace code for nets [i] based on (17, 55, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
    - base change [i] based on digital (9, 48, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(72, 72+31, 2184)-Net over F₈ — Digital

Digital (72, 103, 2184)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+31, 1269630)-Net in Base 8 — Upper bound on s

There is no (72, 103, 1269631)-net in base 8, because

1 times m-reduction [i] would yield (72, 102, 1269631)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 130 370901 333530 254040 840016 359337 622069 037669 093121 552212 518937 200221 887261 534411 661365 910936 > 8¹⁰² [i]