MinT - Best Known (64−39, 64, s)-Nets in Base 27

Best Known (64−39, 64, s)-Nets in Base 27

(64−39, 64, 114)-Net over F₂₇ — Constructive and digital

Digital (25, 64, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 64, 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]

(64−39, 64, 172)-Net in Base 27 — Constructive

(25, 64, 172)-net in base 27, using

8 times m-reduction [i] based on (25, 72, 172)-net in base 27, using
- base change [i] based on digital (7, 54, 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]

(64−39, 64, 208)-Net over F₂₇ — Digital

Digital (25, 64, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 64, 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]

(64−39, 64, 244)-Net in Base 27

(25, 64, 244)-net in base 27, using

base change [i] based on digital (9, 48, 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]

(64−39, 64, 16986)-Net in Base 27 — Upper bound on s

There is no (25, 64, 16987)-net in base 27, because

1 times m-reduction [i] would yield (25, 63, 16987)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 500862 391203 007064 550163 102138 452507 423877 467089 066701 799069 449503 872125 103129 094177 579779 > 27⁶³ [i]