MinT - Best Known (24, 65, s)-Nets in Base 27

Best Known (24, 65, s)-Nets in Base 27

(24, 65, 114)-Net over F₂₇ — Constructive and digital

Digital (24, 65, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 65, 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]

(24, 65, 172)-Net in Base 27 — Constructive

(24, 65, 172)-net in base 27, using

3 times m-reduction [i] based on (24, 68, 172)-net in base 27, using
- base change [i] based on digital (7, 51, 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]

(24, 65, 208)-Net over F₂₇ — Digital

Digital (24, 65, 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]

(24, 65, 12143)-Net in Base 27 — Upper bound on s

There is no (24, 65, 12144)-net in base 27, because

1 times m-reduction [i] would yield (24, 64, 12144)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 40 543194 390266 907353 583420 257992 393340 717646 188220 708083 840417 105539 727210 467723 426428 625537 > 27⁶⁴ [i]