MinT - Best Known (55, 55+44, s)-Nets in Base 32

Best Known (55, 55+44, s)-Nets in Base 32

(55, 55+44, 272)-Net over F₃₂ — Constructive and digital

Digital (55, 99, 272)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 19, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, 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, 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, 51, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(55, 55+44, 513)-Net in Base 32 — Constructive

(55, 99, 513)-net in base 32, using

t-expansion [i] based on (46, 99, 513)-net in base 32, using
- 9 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]

(55, 55+44, 1612)-Net over F₃₂ — Digital

Digital (55, 99, 1612)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(55, 55+44, 1732460)-Net in Base 32 — Upper bound on s

There is no (55, 99, 1732461)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 102293 670767 193387 594740 898280 848240 568251 795073 797910 910217 923143 887405 931015 228108 356282 977392 978381 025701 283340 785856 474449 321285 291713 107540 789336 > 32⁹⁹ [i]