MinT - Best Known (68−45, 68, s)-Nets in Base 32

Best Known (68−45, 68, s)-Nets in Base 32

(68−45, 68, 120)-Net over F₃₂ — Constructive and digital

Digital (23, 68, 120)-net over F₃₂, using

t-expansion [i] based on digital (11, 68, 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]

(68−45, 68, 185)-Net over F₃₂ — Digital

Digital (23, 68, 185)-net over F₃₂, using

t-expansion [i] based on digital (21, 68, 185)-net over F₃₂, using
- net from sequence [i] based on digital (21, 184)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 21 and N(F) ≥ 185, using
    - a curve from the manYPoints database [i]

(68−45, 68, 192)-Net in Base 32 — Constructive

(23, 68, 192)-net in base 32, using

2 times m-reduction [i] based on (23, 70, 192)-net in base 32, using
- base change [i] based on digital (3, 50, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(68−45, 68, 225)-Net in Base 32

(23, 68, 225)-net in base 32, using

10 times m-reduction [i] based on (23, 78, 225)-net in base 32, using
- base change [i] based on digital (10, 65, 225)-net over F₆₄, using
  - net from sequence [i] based on digital (10, 224)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
      - a curve from the manYPoints database [i]

(68−45, 68, 11192)-Net in Base 32 — Upper bound on s

There is no (23, 68, 11193)-net in base 32, because

1 times m-reduction [i] would yield (23, 67, 11193)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 70077 010137 254366 707010 558238 250179 721693 530290 076966 930450 642653 380944 473049 841762 646026 037902 611516 > 32⁶⁷ [i]