MinT - Best Known (98−51, 98, s)-Nets in Base 32

Best Known (98−51, 98, s)-Nets in Base 32

(98−51, 98, 240)-Net over F₃₂ — Constructive and digital

Digital (47, 98, 240)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (11, 36, 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, 62, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(98−51, 98, 513)-Net in Base 32 — Constructive

(47, 98, 513)-net in base 32, using

t-expansion [i] based on (46, 98, 513)-net in base 32, using
- 10 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]

(98−51, 98, 567)-Net over F₃₂ — Digital

Digital (47, 98, 567)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(98−51, 98, 227102)-Net in Base 32 — Upper bound on s

There is no (47, 98, 227103)-net in base 32, because

1 times m-reduction [i] would yield (47, 97, 227103)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 99 903493 444182 372795 737811 795982 460386 217376 026253 286697 867782 155552 776189 064882 559774 989536 300328 826465 322826 782160 729840 475185 603287 079510 626966 > 32⁹⁷ [i]