MinT - Best Known (105−45, 105, s)-Nets in Base 32

Best Known (105−45, 105, s)-Nets in Base 32

(105−45, 105, 300)-Net over F₃₂ — Constructive and digital

Digital (60, 105, 300)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 22, 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]
2. digital (7, 29, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
3. digital (9, 54, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(105−45, 105, 513)-Net in Base 32 — Constructive

(60, 105, 513)-net in base 32, using

t-expansion [i] based on (46, 105, 513)-net in base 32, using
- 3 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]

(105−45, 105, 2197)-Net over F₃₂ — Digital

Digital (60, 105, 2197)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(105−45, 105, 3808417)-Net in Base 32 — Upper bound on s

There is no (60, 105, 3808418)-net in base 32, because

1 times m-reduction [i] would yield (60, 104, 3808418)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 432418 193318 615691 070958 140331 373692 582007 947412 233881 925268 856653 294691 920425 921721 014145 638118 749500 162473 331837 666537 883830 457270 741474 824492 202945 440656 > 32¹⁰⁴ [i]