MinT - Best Known (108−49, 108, s)-Nets in Base 32

Best Known (108−49, 108, s)-Nets in Base 32

(108−49, 108, 272)-Net over F₃₂ — Constructive and digital

Digital (59, 108, 272)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 21, 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, 31, 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, 56, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(108−49, 108, 513)-Net in Base 32 — Constructive

(59, 108, 513)-net in base 32, using

t-expansion [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]

(108−49, 108, 1496)-Net over F₃₂ — Digital

Digital (59, 108, 1496)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(108−49, 108, 1623521)-Net in Base 32 — Upper bound on s

There is no (59, 108, 1623522)-net in base 32, because

1 times m-reduction [i] would yield (59, 107, 1623522)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 112473 459875 974270 697807 709176 490597 666838 315524 535220 307626 121947 839802 737064 644746 182017 724184 413719 813557 027576 476767 534451 621941 962280 587024 449552 205416 812628 > 32¹⁰⁷ [i]