MinT - Best Known (101−43, 101, s)-Nets in Base 32

Best Known (101−43, 101, s)-Nets in Base 32

(101−43, 101, 300)-Net over F₃₂ — Constructive and digital

Digital (58, 101, 300)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 21, 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]
2. digital (7, 28, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
3. digital (9, 52, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(101−43, 101, 513)-Net in Base 32 — Constructive

(58, 101, 513)-net in base 32, using

t-expansion [i] based on (46, 101, 513)-net in base 32, using
- 7 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]

(101−43, 101, 2239)-Net over F₃₂ — Digital

Digital (58, 101, 2239)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(101−43, 101, 4116333)-Net in Base 32 — Upper bound on s

There is no (58, 101, 4116334)-net in base 32, because

1 times m-reduction [i] would yield (58, 100, 4116334)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 273392 228222 501305 692640 074347 133311 285072 536523 843498 605778 186585 039802 682673 722160 789638 254865 627228 137691 178090 728847 465741 190138 168071 527659 439470 > 32¹⁰⁰ [i]