MinT - Best Known (28, 28+45, s)-Nets in Base 32

Best Known (28, 28+45, s)-Nets in Base 32

(28, 28+45, 128)-Net over F₃₂ — Constructive and digital

Digital (28, 73, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 25, 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, 48, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(28, 28+45, 257)-Net in Base 32 — Constructive

(28, 73, 257)-net in base 32, using

32¹ times duplication [i] based on (27, 72, 257)-net in base 32, using
- base change [i] based on digital (0, 45, 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]

(28, 28+45, 257)-Net over F₃₂ — Digital

Digital (28, 73, 257)-net over F₃₂, using

net from sequence [i] based on digital (28, 256)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 28 and N(F) ≥ 257, using
  - a curve from the manYPoints database [i]

(28, 28+45, 258)-Net in Base 32

(28, 73, 258)-net in base 32, using

5 times m-reduction [i] based on (28, 78, 258)-net in base 32, using
- base change [i] based on digital (15, 65, 258)-net over F₆₄, using
  - net from sequence [i] based on digital (15, 257)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 15 and N(F) ≥ 258, using
      - a curve from the manYPoints database [i]

(28, 28+45, 24617)-Net in Base 32 — Upper bound on s

There is no (28, 73, 24618)-net in base 32, because

1 times m-reduction [i] would yield (28, 72, 24618)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 350177 732704 718283 372557 277411 393246 680997 904726 983853 605222 996930 336396 242229 920342 695616 705560 464383 202936 > 32⁷² [i]