MinT - Best Known (30, 30+59, s)-Nets in Base 27

Best Known (30, 30+59, s)-Nets in Base 27

(30, 30+59, 114)-Net over F₂₇ — Constructive and digital

Digital (30, 89, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 89, 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]

(30, 30+59, 172)-Net in Base 27 — Constructive

(30, 89, 172)-net in base 27, using

3 times m-reduction [i] based on (30, 92, 172)-net in base 27, using
- base change [i] based on digital (7, 69, 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]

(30, 30+59, 208)-Net over F₂₇ — Digital

Digital (30, 89, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 89, 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]

(30, 30+59, 9884)-Net in Base 27 — Upper bound on s

There is no (30, 89, 9885)-net in base 27, because

1 times m-reduction [i] would yield (30, 88, 9885)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 914637 753381 714329 976362 369638 392413 015525 480542 527738 080155 590052 204097 448405 066768 992162 460694 412992 954375 357697 584349 736387 > 27⁸⁸ [i]