MinT - Best Known (96−51, 96, s)-Nets in Base 32

Best Known (96−51, 96, s)-Nets in Base 32

(96−51, 96, 224)-Net over F₃₂ — Constructive and digital

Digital (45, 96, 224)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 34, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, 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, 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]

(96−51, 96, 490)-Net over F₃₂ — Digital

Digital (45, 96, 490)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(96−51, 96, 513)-Net in Base 32 — Constructive

(45, 96, 513)-net in base 32, using

6 times m-reduction [i] based on (45, 102, 513)-net in base 32, using
- base change [i] based on digital (28, 85, 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]

(96−51, 96, 172108)-Net in Base 32 — Upper bound on s

There is no (45, 96, 172109)-net in base 32, because

1 times m-reduction [i] would yield (45, 95, 172109)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 97564 997960 053983 900376 404006 132663 156466 213521 767802 474677 612686 380048 557603 181281 312704 306661 972573 705340 765887 974146 576034 155951 584463 324816 > 32⁹⁵ [i]