MinT - Best Known (72, 72+38, s)-Nets in Base 8

Best Known (72, 72+38, s)-Nets in Base 8

(72, 72+38, 368)-Net over F₈ — Constructive and digital

Digital (72, 110, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (52, 90, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 45, 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+38, 576)-Net in Base 8 — Constructive

(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+38, 1032)-Net over F₈ — Digital

Digital (72, 110, 1032)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+38, 191649)-Net in Base 8 — Upper bound on s

There is no (72, 110, 191650)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2187 460034 936533 705016 166323 791888 600362 679308 193309 405194 269582 138655 130413 608939 456262 776403 140816 > 8¹¹⁰ [i]