MinT - Best Known (32, 32+48, s)-Nets in Base 27

Best Known (32, 32+48, s)-Nets in Base 27

(32, 32+48, 128)-Net over F₂₇ — Constructive and digital

Digital (32, 80, 128)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 28, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (4, 52, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(32, 32+48, 172)-Net in Base 27 — Constructive

(32, 80, 172)-net in base 27, using

20 times m-reduction [i] based on (32, 100, 172)-net in base 27, using
- base change [i] based on digital (7, 75, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(32, 32+48, 208)-Net over F₂₇ — Digital

Digital (32, 80, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 80, 208)-net over F₂₇, using
- net from sequence [i] based on digital (24, 207)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 24 and N(F) ≥ 208, using
    - an algebraic function field by Beelen and Pellikaan [i]

(32, 32+48, 298)-Net in Base 27

(32, 80, 298)-net in base 27, using

base change [i] based on digital (12, 60, 298)-net over F₈₁, using
- net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
    - a curve from the manYPoints database [i]

(32, 32+48, 22251)-Net in Base 27 — Upper bound on s

There is no (32, 80, 22252)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 230938 194856 723845 529086 610375 082791 093885 455888 705254 462214 807312 509614 798765 552233 781832 625350 008606 356424 442433 > 27⁸⁰ [i]