MinT - Best Known (68, 68+42, s)-Nets in Base 32

Best Known (68, 68+42, s)-Nets in Base 32

(68, 68+42, 360)-Net over F₃₂ — Constructive and digital

Digital (68, 110, 360)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (11, 25, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]
2. digital (11, 32, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)
3. digital (11, 53, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(68, 68+42, 589)-Net in Base 32 — Constructive

(68, 110, 589)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (5, 26, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. (42, 84, 513)-net in base 32, using
  - base change [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]

(68, 68+42, 5706)-Net over F₃₂ — Digital

Digital (68, 110, 5706)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+42, large)-Net in Base 32 — Upper bound on s

There is no (68, 110, large)-net in base 32, because

40 times m-reduction [i] would yield (68, 70, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]