MinT - Best Known (85−55, 85, s)-Nets in Base 27

Best Known (85−55, 85, s)-Nets in Base 27

(85−55, 85, 114)-Net over F₂₇ — Constructive and digital

Digital (30, 85, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 85, 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]

(85−55, 85, 172)-Net in Base 27 — Constructive

(30, 85, 172)-net in base 27, using

7 times m-reduction [i] based on (30, 92, 172)-net in base 27, using
- base change [i] based on digital (7, 69, 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]

(85−55, 85, 208)-Net over F₂₇ — Digital

Digital (30, 85, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 85, 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]

(85−55, 85, 226)-Net in Base 27

(30, 85, 226)-net in base 27, using

3 times m-reduction [i] based on (30, 88, 226)-net in base 27, using
- base change [i] based on digital (8, 66, 226)-net over F₈₁, using
  - net from sequence [i] based on digital (8, 225)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 8 and N(F) ≥ 226, using
      - a curve from the manYPoints database [i]

(85−55, 85, 11913)-Net in Base 27 — Upper bound on s

There is no (30, 85, 11914)-net in base 27, because

1 times m-reduction [i] would yield (30, 84, 11914)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 716555 231520 615127 417242 382999 862999 882836 416967 278052 094030 594521 237067 090891 107041 351076 727004 066489 707419 644604 953137 > 27⁸⁴ [i]