MinT - Best Known (82−55, 82, s)-Nets in Base 27

Best Known (82−55, 82, s)-Nets in Base 27

(82−55, 82, 114)-Net over F₂₇ — Constructive and digital

Digital (27, 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]

(82−55, 82, 160)-Net in Base 27 — Constructive

(27, 82, 160)-net in base 27, using

6 times m-reduction [i] based on (27, 88, 160)-net in base 27, using
- base change [i] based on digital (5, 66, 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]

(82−55, 82, 208)-Net over F₂₇ — Digital

Digital (27, 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]

(82−55, 82, 8256)-Net in Base 27 — Upper bound on s

There is no (27, 82, 8257)-net in base 27, because

1 times m-reduction [i] would yield (27, 81, 8257)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 87 379809 653044 748251 753674 631202 937759 152037 359940 119062 710878 139170 820573 619448 931003 394303 959459 111960 362589 281731 > 27⁸¹ [i]