MinT - Best Known (102−55, 102, s)-Nets in Base 32

Best Known (102−55, 102, s)-Nets in Base 32

(102−55, 102, 224)-Net over F₃₂ — Constructive and digital

Digital (47, 102, 224)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 36, 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]
2. digital (11, 66, 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]

(102−55, 102, 468)-Net over F₃₂ — Digital

Digital (47, 102, 468)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(102−55, 102, 513)-Net in Base 32 — Constructive

(47, 102, 513)-net in base 32, using

t-expansion [i] based on (46, 102, 513)-net in base 32, using
- 6 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]

(102−55, 102, 150446)-Net in Base 32 — Upper bound on s

There is no (47, 102, 150447)-net in base 32, because

1 times m-reduction [i] would yield (47, 101, 150447)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 104 765755 855921 806614 065022 902710 719023 806344 906254 930741 191846 325886 971521 393456 346267 549784 015120 110738 133407 225134 874099 097643 018891 439143 139108 179912 > 32¹⁰¹ [i]