MinT - Best Known (99−51, 99, s)-Nets in Base 32

Best Known (99−51, 99, s)-Nets in Base 32

(99−51, 99, 240)-Net over F₃₂ — Constructive and digital

Digital (48, 99, 240)-net over F₃₂, using

1 times m-reduction [i] based on digital (48, 100, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 37, 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, 63, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(99−51, 99, 513)-Net in Base 32 — Constructive

(48, 99, 513)-net in base 32, using

t-expansion [i] based on (46, 99, 513)-net in base 32, using
- 9 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]

(99−51, 99, 610)-Net over F₃₂ — Digital

Digital (48, 99, 610)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(99−51, 99, 260873)-Net in Base 32 — Upper bound on s

There is no (48, 99, 260874)-net in base 32, because

1 times m-reduction [i] would yield (48, 98, 260874)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3196 683025 663250 589729 583425 992091 819774 236461 507133 023041 569379 551737 799001 353977 569578 965968 266946 514451 832089 501769 727803 341593 960089 094651 475536 > 32⁹⁸ [i]