MinT - Best Known (88−39, 88, s)-Nets in Base 32

Best Known (88−39, 88, s)-Nets in Base 32

(88−39, 88, 260)-Net over F₃₂ — Constructive and digital

Digital (49, 88, 260)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 16, 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, 46, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(88−39, 88, 513)-Net in Base 32 — Constructive

(49, 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]

(88−39, 88, 1502)-Net over F₃₂ — Digital

Digital (49, 88, 1502)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(88−39, 88, 1994592)-Net in Base 32 — Upper bound on s

There is no (49, 88, 1994593)-net in base 32, because

1 times m-reduction [i] would yield (49, 87, 1994593)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 88725 580704 358843 072727 316711 231501 839985 885487 835598 108331 811268 492767 278208 716031 393854 916769 827250 684840 583656 067957 508587 307552 > 32⁸⁷ [i]