MinT - Best Known (108−59, 108, s)-Nets in Base 32

Best Known (108−59, 108, s)-Nets in Base 32

(108−59, 108, 224)-Net over F₃₂ — Constructive and digital

Digital (49, 108, 224)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 38, 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, 70, 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]

(108−59, 108, 453)-Net over F₃₂ — Digital

Digital (49, 108, 453)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(108−59, 108, 513)-Net in Base 32 — Constructive

(49, 108, 513)-net in base 32, using

t-expansion [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]

(108−59, 108, 134646)-Net in Base 32 — Upper bound on s

There is no (49, 108, 134647)-net in base 32, because

1 times m-reduction [i] would yield (49, 107, 134647)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 112473 700110 527296 367280 283724 145533 335150 462579 263366 121204 824190 738774 116427 788292 528704 251875 488757 456983 446401 985244 047025 068608 048179 380974 876411 179676 646456 > 32¹⁰⁷ [i]