MinT - Best Known (125−29, 125, s)-Nets in Base 8

Best Known (125−29, 125, s)-Nets in Base 8

(125−29, 125, 1026)-Net over F₈ — Constructive and digital

Digital (96, 125, 1026)-net over F₈, using

11 times m-reduction [i] based on digital (96, 136, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 68, 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]

(125−29, 125, 1034)-Net in Base 8 — Constructive

(96, 125, 1034)-net in base 8, using

8¹ times duplication [i] based on (95, 124, 1034)-net in base 8, using
- base change [i] based on digital (64, 93, 1034)-net over F₁₆, using
  - 16¹ times duplication [i] based on digital (63, 92, 1034)-net over F₁₆, using
    - (u,Â u+v)-construction [i] based on
      1. digital (16, 30, 516)-net over F₁₆, using
        trace code for nets [i] based on digital (1, 15, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
      2. digital (33, 62, 518)-net over F₁₆, using
        trace code for nets [i] based on digital (2, 31, 259)-net over F₂₅₆, using
        net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(125−29, 125, 17374)-Net over F₈ — Digital

Digital (96, 125, 17374)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−29, 125, large)-Net in Base 8 — Upper bound on s

There is no (96, 125, large)-net in base 8, because

27 times m-reduction [i] would yield (96, 98, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]