MinT - Best Known (82, 82+41, s)-Nets in Base 8

Best Known (82, 82+41, s)-Nets in Base 8

(82, 82+41, 388)-Net over F₈ — Constructive and digital

Digital (82, 123, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 27, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (55, 96, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 48, 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+41, 576)-Net in Base 8 — Constructive

(82, 123, 576)-net in base 8, using

t-expansion [i] based on (81, 123, 576)-net in base 8, using
- 3 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+41, 1368)-Net over F₈ — Digital

Digital (82, 123, 1368)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+41, 382862)-Net in Base 8 — Upper bound on s

There is no (82, 123, 382863)-net in base 8, because

1 times m-reduction [i] would yield (82, 122, 382863)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 150 308824 714104 277363 347000 592897 915898 042063 928583 949392 651662 439584 689553 598542 473389 868949 085125 393413 164984 > 8¹²² [i]