MinT - Best Known (50, 50+53, s)-Nets in Base 32

Best Known (50, 50+53, s)-Nets in Base 32

(50, 50+53, 240)-Net over F₃₂ — Constructive and digital

Digital (50, 103, 240)-net over F₃₂, using

3 times m-reduction [i] based on digital (50, 106, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 39, 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, 67, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(50, 50+53, 513)-Net in Base 32 — Constructive

(50, 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]

(50, 50+53, 633)-Net over F₃₂ — Digital

Digital (50, 103, 633)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+53, 273356)-Net in Base 32 — Upper bound on s

There is no (50, 103, 273357)-net in base 32, because

1 times m-reduction [i] would yield (50, 102, 273357)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3352 132445 426784 454673 289360 542998 291875 400450 423763 081058 571829 576969 976255 448129 424948 964469 597240 927960 249739 160930 339221 830029 685905 534330 916320 208888 > 32¹⁰² [i]