MinT - Best Known (29, 29+57, s)-Nets in Base 27

Best Known (29, 29+57, s)-Nets in Base 27

(29, 29+57, 114)-Net over F₂₇ — Constructive and digital

Digital (29, 86, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 86, 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]

(29, 29+57, 172)-Net in Base 27 — Constructive

(29, 86, 172)-net in base 27, using

2 times m-reduction [i] based on (29, 88, 172)-net in base 27, using
- base change [i] based on digital (7, 66, 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]

(29, 29+57, 208)-Net over F₂₇ — Digital

Digital (29, 86, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 86, 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]

(29, 29+57, 9606)-Net in Base 27 — Upper bound on s

There is no (29, 86, 9607)-net in base 27, because

1 times m-reduction [i] would yield (29, 85, 9607)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 46 344887 397588 371401 048806 662910 017132 079478 470567 447872 300280 678083 930716 264930 437356 452958 261438 163993 660949 107606 241065 > 27⁸⁵ [i]