MinT - Best Known (78−48, 78, s)-Nets in Base 32

Best Known (78−48, 78, s)-Nets in Base 32

(78−48, 78, 128)-Net over F₃₂ — Constructive and digital

Digital (30, 78, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 27, 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, 51, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(78−48, 78, 257)-Net in Base 32 — Constructive

(30, 78, 257)-net in base 32, using

2 times m-reduction [i] based on (30, 80, 257)-net in base 32, using
- base change [i] based on digital (0, 50, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]

(78−48, 78, 273)-Net over F₃₂ — Digital

Digital (30, 78, 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]

(78−48, 78, 24633)-Net in Base 32 — Upper bound on s

There is no (30, 78, 24634)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2523 510008 020452 291262 948443 081223 656830 595245 871214 798982 254677 648492 644838 944978 673911 331032 356270 199567 030687 308931 > 32⁷⁸ [i]