MinT - Best Known (106−59, 106, s)-Nets in Base 32

Best Known (106−59, 106, s)-Nets in Base 32

(106−59, 106, 218)-Net over F₃₂ — Constructive and digital

Digital (47, 106, 218)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 36, 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, 70, 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]

(106−59, 106, 399)-Net over F₃₂ — Digital

Digital (47, 106, 399)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(106−59, 106, 513)-Net in Base 32 — Constructive

(47, 106, 513)-net in base 32, using

t-expansion [i] based on (46, 106, 513)-net in base 32, using
- 2 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]

(106−59, 106, 106018)-Net in Base 32 — Upper bound on s

There is no (47, 106, 106019)-net in base 32, because

1 times m-reduction [i] would yield (47, 105, 106019)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 109 863084 123050 610979 862048 684083 432635 413712 488344 116978 091224 015394 775596 128012 048577 324545 788326 921814 119344 074568 657925 752024 799595 570142 545701 803261 008144 > 32¹⁰⁵ [i]