MinT - Best Known (69−44, 69, s)-Nets in Base 27

Best Known (69−44, 69, s)-Nets in Base 27

(69−44, 69, 114)-Net over F₂₇ — Constructive and digital

Digital (25, 69, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 69, 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]

(69−44, 69, 172)-Net in Base 27 — Constructive

(25, 69, 172)-net in base 27, using

3 times m-reduction [i] based on (25, 72, 172)-net in base 27, using
- base change [i] based on digital (7, 54, 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]

(69−44, 69, 208)-Net over F₂₇ — Digital

Digital (25, 69, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 69, 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]

(69−44, 69, 10732)-Net in Base 27 — Upper bound on s

There is no (25, 69, 10733)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 581 573288 368114 564015 034335 487577 882300 820054 124809 196235 500481 838879 982684 529569 893452 733915 939329 > 27⁶⁹ [i]