MinT - Best Known (82, 82+34, s)-Nets in Base 8

Best Known (82, 82+34, s)-Nets in Base 8

(82, 82+34, 484)-Net over F₈ — Constructive and digital

Digital (82, 116, 484)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (17, 34, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 17, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. digital (48, 82, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 41, 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+34, 576)-Net in Base 8 — Constructive

(82, 116, 576)-net in base 8, using

t-expansion [i] based on (81, 116, 576)-net in base 8, using
- 10 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+34, 2827)-Net over F₈ — Digital

Digital (82, 116, 2827)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+34, 1489721)-Net in Base 8 — Upper bound on s

There is no (82, 116, 1489722)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 573 375038 631848 896492 157295 659445 683035 809108 585741 119810 904803 806697 537872 208340 613766 225600 127058 915631 > 8¹¹⁶ [i]