MinT - Best Known (25, 25+57, s)-Nets in Base 27

Best Known (25, 25+57, s)-Nets in Base 27

(25, 25+57, 114)-Net over F₂₇ — Constructive and digital

Digital (25, 82, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 82, 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]

(25, 25+57, 150)-Net in Base 27 — Constructive

(25, 82, 150)-net in base 27, using

2 times m-reduction [i] based on (25, 84, 150)-net in base 27, using
- base change [i] based on digital (4, 63, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(25, 25+57, 208)-Net over F₂₇ — Digital

Digital (25, 82, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 82, 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]

(25, 25+57, 5993)-Net in Base 27 — Upper bound on s

There is no (25, 82, 5994)-net in base 27, because

1 times m-reduction [i] would yield (25, 81, 5994)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 87 237330 601700 175078 720298 918461 840722 288543 620828 756212 433892 844264 191610 307962 766102 887738 195546 892393 633104 473113 > 27⁸¹ [i]