MinT - Best Known (110−57, 110, s)-Nets in Base 32

Best Known (110−57, 110, s)-Nets in Base 32

(110−57, 110, 240)-Net over F₃₂ — Constructive and digital

Digital (53, 110, 240)-net over F₃₂, using

t-expansion [i] based on digital (51, 110, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 40, 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]
  2. digital (11, 70, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(110−57, 110, 513)-Net in Base 32 — Constructive

(53, 110, 513)-net in base 32, using

32² times duplication [i] based on (51, 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]

(110−57, 110, 638)-Net over F₃₂ — Digital

Digital (53, 110, 638)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−57, 110, 263604)-Net in Base 32 — Upper bound on s

There is no (53, 110, 263605)-net in base 32, because

1 times m-reduction [i] would yield (53, 109, 263605)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 115 173623 004606 279370 048974 582814 520181 792680 677228 664155 803423 160620 350783 317878 286435 219177 320164 634883 888675 675054 858237 293240 338330 413500 882513 934947 903736 180596 > 32¹⁰⁹ [i]