MinT - Best Known (82, 82+43, s)-Nets in Base 8

Best Known (82, 82+43, s)-Nets in Base 8

(82, 82+43, 379)-Net over F₈ — Constructive and digital

Digital (82, 125, 379)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 25, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (57, 100, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 50, 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]

(82, 82+43, 576)-Net in Base 8 — Constructive

(82, 125, 576)-net in base 8, using

t-expansion [i] based on (81, 125, 576)-net in base 8, using
- 1 times m-reduction [i] based on (81, 126, 576)-net in base 8, using
  - trace code for nets [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 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]

(82, 82+43, 1170)-Net over F₈ — Digital

Digital (82, 125, 1170)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+43, 266627)-Net in Base 8 — Upper bound on s

There is no (82, 125, 266628)-net in base 8, because

1 times m-reduction [i] would yield (82, 124, 266628)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 9620 118898 820671 452127 370677 159981 099398 548796 738140 119211 736256 853554 052416 999509 727705 805059 716708 153360 840032 > 8¹²⁴ [i]