MinT - Best Known (50, 50+51, s)-Nets in Base 32

Best Known (50, 50+51, s)-Nets in Base 32

(50, 50+51, 240)-Net over F₃₂ — Constructive and digital

Digital (50, 101, 240)-net over F₃₂, using

5 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+51, 513)-Net in Base 32 — Constructive

(50, 101, 513)-net in base 32, using

t-expansion [i] based on (46, 101, 513)-net in base 32, using
- 7 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+51, 705)-Net over F₃₂ — Digital

Digital (50, 101, 705)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+51, 344229)-Net in Base 32 — Upper bound on s

There is no (50, 101, 344230)-net in base 32, because

1 times m-reduction [i] would yield (50, 100, 344230)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 273546 436805 118350 979954 111819 375275 569804 683283 520044 469014 987686 379680 014663 273216 593483 152298 659299 644105 064970 937850 849575 098916 316250 539342 549276 > 32¹⁰⁰ [i]