MinT - Best Known (82−51, 82, s)-Nets in Base 32

Best Known (82−51, 82, s)-Nets in Base 32

(82−51, 82, 128)-Net over F₃₂ — Constructive and digital

Digital (31, 82, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 28, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (3, 54, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(82−51, 82, 216)-Net in Base 32 — Constructive

(31, 82, 216)-net in base 32, using

9 times m-reduction [i] based on (31, 91, 216)-net in base 32, using
- base change [i] based on digital (5, 65, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(82−51, 82, 273)-Net over F₃₂ — Digital

Digital (31, 82, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 82, 273)-net over F₃₂, using
- net from sequence [i] based on digital (30, 272)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273, using
    - a curve from the manYPoints database [i]

(82−51, 82, 24701)-Net in Base 32 — Upper bound on s

There is no (31, 82, 24702)-net in base 32, because

1 times m-reduction [i] would yield (31, 81, 24702)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 82 665894 727193 713007 518736 334275 794104 874793 532744 592581 937094 574140 206336 990898 548925 476615 509575 904761 679982 345756 521095 > 32⁸¹ [i]