MinT - Best Known (82−53, 82, s)-Nets in Base 27

Best Known (82−53, 82, s)-Nets in Base 27

(82−53, 82, 114)-Net over F₂₇ — Constructive and digital

Digital (29, 82, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 82, 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]

(82−53, 82, 172)-Net in Base 27 — Constructive

(29, 82, 172)-net in base 27, using

6 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]

(82−53, 82, 208)-Net over F₂₇ — Digital

Digital (29, 82, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 82, 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]

(82−53, 82, 226)-Net in Base 27

(29, 82, 226)-net in base 27, using

2 times m-reduction [i] based on (29, 84, 226)-net in base 27, using
- base change [i] based on digital (8, 63, 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]

(82−53, 82, 11669)-Net in Base 27 — Upper bound on s

There is no (29, 82, 11670)-net in base 27, because

1 times m-reduction [i] would yield (29, 81, 11670)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 87 213086 426461 264130 671501 340161 909462 627000 235656 248624 771705 797847 821529 336495 619273 939806 699026 591365 495449 001533 > 27⁸¹ [i]