MinT - Best Known (151−53, 151, s)-Nets in Base 8

Best Known (151−53, 151, s)-Nets in Base 8

(151−53, 151, 382)-Net over F₈ — Constructive and digital

Digital (98, 151, 382)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 31, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (67, 120, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 60, 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]

(151−53, 151, 576)-Net in Base 8 — Constructive

(98, 151, 576)-net in base 8, using

t-expansion [i] based on (97, 151, 576)-net in base 8, using
- 3 times m-reduction [i] based on (97, 154, 576)-net in base 8, using
  - trace code for nets [i] based on (20, 77, 288)-net in base 64, using
    - base change [i] based on digital (9, 66, 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]

(151−53, 151, 1237)-Net over F₈ — Digital

Digital (98, 151, 1237)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(151−53, 151, 244512)-Net in Base 8 — Upper bound on s

There is no (98, 151, 244513)-net in base 8, because

1 times m-reduction [i] would yield (98, 150, 244513)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2907 565858 172465 653038 297791 503052 135123 350525 738650 283470 847595 181486 678178 863325 224916 863484 736952 442171 590313 948042 036137 454887 159376 > 8¹⁵⁰ [i]