MinT - Best Known (100−55, 100, s)-Nets in Base 32

Best Known (100−55, 100, s)-Nets in Base 32

(100−55, 100, 218)-Net over F₃₂ — Constructive and digital

Digital (45, 100, 218)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 34, 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 (11, 66, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(100−55, 100, 409)-Net over F₃₂ — Digital

Digital (45, 100, 409)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(100−55, 100, 513)-Net in Base 32 — Constructive

(45, 100, 513)-net in base 32, using

2 times m-reduction [i] based on (45, 102, 513)-net in base 32, using
- base change [i] based on digital (28, 85, 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]

(100−55, 100, 116379)-Net in Base 32 — Upper bound on s

There is no (45, 100, 116380)-net in base 32, because

1 times m-reduction [i] would yield (45, 99, 116380)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 102304 780860 739677 620862 086704 160530 692957 617757 089009 811277 523419 043548 658163 974814 205103 185404 766833 002999 529657 919525 565120 173166 623358 597216 442487 > 32⁹⁹ [i]