MinT - Best Known (108−50, 108, s)-Nets in Base 32

Best Known (108−50, 108, s)-Nets in Base 32

(108−50, 108, 260)-Net over F₃₂ — Constructive and digital

Digital (58, 108, 260)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 19, 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, 32, 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, 57, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(108−50, 108, 513)-Net in Base 32 — Constructive

(58, 108, 513)-net in base 32, using

t-expansion [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]

(108−50, 108, 1305)-Net over F₃₂ — Digital

Digital (58, 108, 1305)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(108−50, 108, 1043534)-Net in Base 32 — Upper bound on s

There is no (58, 108, 1043535)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 599195 231182 930466 294609 466059 077737 177534 141527 193914 440174 781107 626538 026897 783517 742784 470392 143159 374023 088409 555956 932609 226169 325658 969415 600411 786045 051608 > 32¹⁰⁸ [i]