MinT - Best Known (39, 39+70, s)-Nets in Base 27

Best Known (39, 39+70, s)-Nets in Base 27

(39, 39+70, 114)-Net over F₂₇ — Constructive and digital

Digital (39, 109, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 109, 114)-net over F₂₇, using
- net from sequence [i] based on digital (23, 113)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 23 and N(F) ≥ 114, using
    - an explicitly constructive algebraic function field [i]

(39, 39+70, 172)-Net in Base 27 — Constructive

(39, 109, 172)-net in base 27, using

27¹ times duplication [i] based on (38, 108, 172)-net in base 27, using
- t-expansion [i] based on (34, 108, 172)-net in base 27, using
  - base change [i] based on digital (7, 81, 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]

(39, 39+70, 271)-Net over F₂₇ — Digital

Digital (39, 109, 271)-net over F₂₇, using

net from sequence [i] based on digital (39, 270)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 39 and N(F) ≥ 271, using
  - a curve from the manYPoints database [i]

(39, 39+70, 15326)-Net in Base 27 — Upper bound on s

There is no (39, 109, 15327)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 044753 509208 737004 035646 755934 617629 308835 394177 375706 695321 470629 123891 370775 913934 507431 334605 320759 960433 616499 306969 943214 606313 046606 613468 601944 077555 > 27¹⁰⁹ [i]