MinT - Best Known (88−57, 88, s)-Nets in Base 27

Best Known (88−57, 88, s)-Nets in Base 27

(88−57, 88, 114)-Net over F₂₇ — Constructive and digital

Digital (31, 88, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 88, 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]

(88−57, 88, 172)-Net in Base 27 — Constructive

(31, 88, 172)-net in base 27, using

8 times m-reduction [i] based on (31, 96, 172)-net in base 27, using
- base change [i] based on digital (7, 72, 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]

(88−57, 88, 208)-Net over F₂₇ — Digital

Digital (31, 88, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 88, 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]

(88−57, 88, 244)-Net in Base 27

(31, 88, 244)-net in base 27, using

base change [i] based on digital (9, 66, 244)-net over F₈₁, using
- net from sequence [i] based on digital (9, 243)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 9 and N(F) ≥ 244, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₈₁ [i]

(88−57, 88, 12160)-Net in Base 27 — Upper bound on s

There is no (31, 88, 12161)-net in base 27, because

1 times m-reduction [i] would yield (31, 87, 12161)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 33792 192795 478390 122299 955620 126893 195597 916320 166504 003622 982623 809064 326843 993668 738311 724534 076537 505484 741614 463563 306129 > 27⁸⁷ [i]