MinT - Best Known (83, 123, s)-Nets in Base 8

Best Known (83, 123, s)-Nets in Base 8

(83, 123, 399)-Net over F₈ — Constructive and digital

Digital (83, 123, 399)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 29, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]
2. digital (54, 94, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 47, 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]

(83, 123, 576)-Net in Base 8 — Constructive

(83, 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]

(83, 123, 1570)-Net over F₈ — Digital

Digital (83, 123, 1570)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(83, 123, 424814)-Net in Base 8 — Upper bound on s

There is no (83, 123, 424815)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1202 501692 565226 124125 232406 599920 876322 519286 551350 924224 200733 053106 257403 716740 562684 759762 667120 415717 538174 > 8¹²³ [i]