MinT - Best Known (88−38, 88, s)-Nets in Base 32

Best Known (88−38, 88, s)-Nets in Base 32

(88−38, 88, 272)-Net over F₃₂ — Constructive and digital

Digital (50, 88, 272)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 17, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (7, 26, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]
3. digital (7, 45, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(88−38, 88, 513)-Net in Base 32 — Constructive

(50, 88, 513)-net in base 32, using

t-expansion [i] based on (46, 88, 513)-net in base 32, using
- 20 times m-reduction [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]

(88−38, 88, 1815)-Net over F₃₂ — Digital

Digital (50, 88, 1815)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(88−38, 88, 2393718)-Net in Base 32 — Upper bound on s

There is no (50, 88, 2393719)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 839231 939326 550318 423630 817921 679259 548986 091412 609101 783779 466226 653164 541922 743359 561638 243654 729636 862456 823710 895759 814320 018836 > 32⁸⁸ [i]