MinT - Best Known (40, 40+29, s)-Nets in Base 32

Best Known (40, 40+29, s)-Nets in Base 32

(40, 40+29, 260)-Net over F₃₂ — Constructive and digital

Digital (40, 69, 260)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 12, 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, 21, 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, 36, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(40, 40+29, 513)-Net in Base 32 — Constructive

(40, 69, 513)-net in base 32, using

3 times m-reduction [i] based on (40, 72, 513)-net in base 32, using
- base change [i] based on digital (28, 60, 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]

(40, 40+29, 1879)-Net over F₃₂ — Digital

Digital (40, 69, 1879)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(40, 40+29, 3988630)-Net in Base 32 — Upper bound on s

There is no (40, 69, 3988631)-net in base 32, because

1 times m-reduction [i] would yield (40, 68, 3988631)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 239752 358981 444470 035624 944218 170636 703460 365707 274837 888308 178809 121122 110328 148044 432990 635315 891178 > 32⁶⁸ [i]