MinT - Best Known (84−54, 84, s)-Nets in Base 27

Best Known (84−54, 84, s)-Nets in Base 27

(84−54, 84, 114)-Net over F₂₇ — Constructive and digital

Digital (30, 84, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 84, 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]

(84−54, 84, 172)-Net in Base 27 — Constructive

(30, 84, 172)-net in base 27, using

8 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]

(84−54, 84, 208)-Net over F₂₇ — Digital

Digital (30, 84, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 84, 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]

(84−54, 84, 244)-Net in Base 27

(30, 84, 244)-net in base 27, using

base change [i] based on digital (9, 63, 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]

(84−54, 84, 11913)-Net in Base 27 — Upper bound on s

There is no (30, 84, 11914)-net in base 27, because

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]