MinT - Best Known (103−55, 103, s)-Nets in Base 32

Best Known (103−55, 103, s)-Nets in Base 32

(103−55, 103, 224)-Net over F₃₂ — Constructive and digital

Digital (48, 103, 224)-net over F₃₂, using

1 times m-reduction [i] based on digital (48, 104, 224)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 37, 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, 67, 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]

(103−55, 103, 501)-Net over F₃₂ — Digital

Digital (48, 103, 501)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(103−55, 103, 513)-Net in Base 32 — Constructive

(48, 103, 513)-net in base 32, using

t-expansion [i] based on (46, 103, 513)-net in base 32, using
- 5 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]

(103−55, 103, 171053)-Net in Base 32 — Upper bound on s

There is no (48, 103, 171054)-net in base 32, because

1 times m-reduction [i] would yield (48, 102, 171054)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3352 208987 426239 399257 123133 653524 139218 578023 979919 289706 136907 071505 169590 750926 857206 486285 989809 240883 839904 772795 827425 087869 037958 148850 440979 271716 > 32¹⁰² [i]