MinT - Best Known (31, 31+63, s)-Nets in Base 27

Best Known (31, 31+63, s)-Nets in Base 27

(31, 31+63, 114)-Net over F₂₇ — Constructive and digital

Digital (31, 94, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 94, 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]

(31, 31+63, 172)-Net in Base 27 — Constructive

(31, 94, 172)-net in base 27, using

2 times m-reduction [i] based on (31, 96, 172)-net in base 27, using
- base change [i] based on digital (7, 72, 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]

(31, 31+63, 208)-Net over F₂₇ — Digital

Digital (31, 94, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 94, 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]

(31, 31+63, 9384)-Net in Base 27 — Upper bound on s

There is no (31, 94, 9385)-net in base 27, because

1 times m-reduction [i] would yield (31, 93, 9385)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13 116384 295661 486921 122522 733030 317349 968232 528262 465907 274365 545361 404582 705250 837430 119509 318209 548627 727263 666115 107968 851926 428531 > 27⁹³ [i]