MinT - Best Known (54, 54+42, s)-Nets in Base 32

Best Known (54, 54+42, s)-Nets in Base 32

(54, 54+42, 272)-Net over F₃₂ — Constructive and digital

Digital (54, 96, 272)-net over F₃₂, using

1 times m-reduction [i] based on digital (54, 97, 272)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (5, 19, 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. digital (7, 28, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]
  3. digital (7, 50, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(54, 54+42, 513)-Net in Base 32 — Constructive

(54, 96, 513)-net in base 32, using

t-expansion [i] based on (46, 96, 513)-net in base 32, using
- 12 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
  - base change [i] based on digital (28, 90, 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]

(54, 54+42, 1762)-Net over F₃₂ — Digital

Digital (54, 96, 1762)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+42, 2127229)-Net in Base 32 — Upper bound on s

There is no (54, 96, 2127230)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 121769 756360 524505 838718 428163 151841 157558 439038 108516 254784 621318 057061 418015 399927 613424 495142 624720 715773 444118 180865 141108 271790 638388 425781 > 32⁹⁶ [i]