MinT - Best Known (31, 31+68, s)-Nets in Base 27

Best Known (31, 31+68, s)-Nets in Base 27

(31, 31+68, 114)-Net over F₂₇ — Constructive and digital

Digital (31, 99, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 99, 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+68, 160)-Net in Base 27 — Constructive

(31, 99, 160)-net in base 27, using

5 times m-reduction [i] based on (31, 104, 160)-net in base 27, using
- base change [i] based on digital (5, 78, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(31, 31+68, 208)-Net over F₂₇ — Digital

Digital (31, 99, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 99, 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+68, 7643)-Net in Base 27 — Upper bound on s

There is no (31, 99, 7644)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 5083 629590 255892 362653 148801 339081 611963 929557 994427 918148 337671 273849 322601 800470 962863 665667 760387 117492 525223 562840 456883 278591 256227 118665 > 27⁹⁹ [i]