MinT - Best Known (39, 39+67, s)-Nets in Base 32

Best Known (39, 39+67, s)-Nets in Base 32

(39, 39+67, 128)-Net over F₃₂ — Constructive and digital

Digital (39, 106, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 36, 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 (3, 70, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(39, 39+67, 216)-Net in Base 32 — Constructive

(39, 106, 216)-net in base 32, using

t-expansion [i] based on (36, 106, 216)-net in base 32, using
- 2 times m-reduction [i] based on (36, 108, 216)-net in base 32, using
  - base change [i] based on (18, 90, 216)-net in base 64, using
    - 1 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
      - base change [i] based on digital (5, 78, 216)-net over F₁₂₈, using
        net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(39, 39+67, 291)-Net over F₃₂ — Digital

Digital (39, 106, 291)-net over F₃₂, using

t-expansion [i] based on digital (38, 106, 291)-net over F₃₂, using
- net from sequence [i] based on digital (38, 290)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 38 and N(F) ≥ 291, using
    - a curve from the manYPoints database [i]

(39, 39+67, 342)-Net in Base 32

(39, 106, 342)-net in base 32, using

t-expansion [i] based on (38, 106, 342)-net in base 32, using
- 2 times m-reduction [i] based on (38, 108, 342)-net in base 32, using
  - base change [i] based on digital (20, 90, 342)-net over F₆₄, using
    - net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
        a curve from the manYPoints database [i]

(39, 39+67, 26110)-Net in Base 32 — Upper bound on s

There is no (39, 106, 26111)-net in base 32, because

1 times m-reduction [i] would yield (39, 105, 26111)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 109 846898 995866 420095 875987 116164 601736 190931 627778 596340 266481 779295 008363 451608 044546 834176 902073 036663 644268 293751 034740 500307 740209 902668 098025 723901 531362 > 32¹⁰⁵ [i]