MinT - Best Known (85−37, 85, s)-Nets in Base 32

Best Known (85−37, 85, s)-Nets in Base 32

(85−37, 85, 260)-Net over F₃₂ — Constructive and digital

Digital (48, 85, 260)-net over F₃₂, using

1 times m-reduction [i] based on digital (48, 86, 260)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (3, 15, 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 (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)

(85−37, 85, 513)-Net in Base 32 — Constructive

(48, 85, 513)-net in base 32, using

t-expansion [i] based on (46, 85, 513)-net in base 32, using
- 23 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]

(85−37, 85, 1667)-Net over F₃₂ — Digital

Digital (48, 85, 1667)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(85−37, 85, 2575135)-Net in Base 32 — Upper bound on s

There is no (48, 85, 2575136)-net in base 32, because

1 times m-reduction [i] would yield (48, 84, 2575136)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 707697 222412 653160 606328 772915 040385 542005 065061 199837 765541 754861 275653 384860 021970 935052 068484 886656 658092 080629 975969 235459 > 32⁸⁴ [i]