MinT - Best Known (138−32, 138, s)-Nets in Base 8

Best Known (138−32, 138, s)-Nets in Base 8

(138−32, 138, 1043)-Net over F₈ — Constructive and digital

Digital (106, 138, 1043)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (88, 120, 1026)-net over F₈, using
  - trace code for nets [i] based on digital (28, 60, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(138−32, 138, 18599)-Net over F₈ — Digital

Digital (106, 138, 18599)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−32, 138, large)-Net in Base 8 — Upper bound on s

There is no (106, 138, large)-net in base 8, because

30 times m-reduction [i] would yield (106, 108, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]