MinT - Best Known (171−42, 171, s)-Nets in Base 8

Best Known (171−42, 171, s)-Nets in Base 8

(171−42, 171, 1072)-Net over F₈ — Constructive and digital

Digital (129, 171, 1072)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 31, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [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 (98, 140, 1026)-net over F₈, using
  - trace code for nets [i] based on digital (28, 70, 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]

(171−42, 171, 13490)-Net over F₈ — Digital

Digital (129, 171, 13490)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(171−42, 171, large)-Net in Base 8 — Upper bound on s

There is no (129, 171, large)-net in base 8, because

40 times m-reduction [i] would yield (129, 131, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]