MinT - Best Known (28, 28+48, s)-Nets in Base 27

Best Known (28, 28+48, s)-Nets in Base 27

(28, 28+48, 114)-Net over F₂₇ — Constructive and digital

Digital (28, 76, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 76, 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]

(28, 28+48, 172)-Net in Base 27 — Constructive

(28, 76, 172)-net in base 27, using

8 times m-reduction [i] based on (28, 84, 172)-net in base 27, using
- base change [i] based on digital (7, 63, 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]

(28, 28+48, 208)-Net over F₂₇ — Digital

Digital (28, 76, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 76, 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]

(28, 28+48, 244)-Net in Base 27

(28, 76, 244)-net in base 27, using

base change [i] based on digital (9, 57, 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]

(28, 28+48, 12841)-Net in Base 27 — Upper bound on s

There is no (28, 76, 12842)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 6 080154 683653 878059 224197 845575 171333 096986 118047 600388 207654 016697 669579 314142 433848 883445 195875 981288 672881 > 27⁷⁶ [i]