MinT - Best Known (93−49, 93, s)-Nets in Base 27

Best Known (93−49, 93, s)-Nets in Base 27

(93−49, 93, 188)-Net over F₂₇ — Constructive and digital

Digital (44, 93, 188)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 34, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]
2. digital (10, 59, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(93−49, 93, 370)-Net in Base 27 — Constructive

(44, 93, 370)-net in base 27, using

t-expansion [i] based on (43, 93, 370)-net in base 27, using
- 15 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(93−49, 93, 428)-Net over F₂₇ — Digital

Digital (44, 93, 428)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(93−49, 93, 115674)-Net in Base 27 — Upper bound on s

There is no (44, 93, 115675)-net in base 27, because

1 times m-reduction [i] would yield (44, 92, 115675)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 484754 879898 084956 798176 138016 455281 570264 661721 246713 835755 517006 407682 310897 184888 055190 698690 240071 391305 024246 589616 537901 729041 > 27⁹² [i]