MinT - Best Known (110−55, 110, s)-Nets in Base 32

Best Known (110−55, 110, s)-Nets in Base 32

(110−55, 110, 240)-Net over F₃₂ — Constructive and digital

Digital (55, 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)

(110−55, 110, 513)-Net in Base 32 — Constructive

(55, 110, 513)-net in base 32, using

32² times duplication [i] based on (53, 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]

(110−55, 110, 804)-Net over F₃₂ — Digital

Digital (55, 110, 804)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−55, 110, 420123)-Net in Base 32 — Upper bound on s

There is no (55, 110, 420124)-net in base 32, because

1 times m-reduction [i] would yield (55, 109, 420124)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 115 176097 398342 637715 433024 296131 409310 547329 150352 260825 696810 030234 651056 484971 062932 501358 260414 319634 891024 602802 123948 613647 024776 272323 498119 610763 199433 520559 > 32¹⁰⁹ [i]