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

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

(48−27, 48, 128)-Net over F₂₇ — Constructive and digital

Digital (21, 48, 128)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 17, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (4, 31, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

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

(21, 48, 172)-net in base 27, using

8 times m-reduction [i] based on (21, 56, 172)-net in base 27, using
- base change [i] based on digital (7, 42, 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]

(48−27, 48, 182)-Net over F₂₇ — Digital

Digital (21, 48, 182)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(48−27, 48, 244)-Net in Base 27

(21, 48, 244)-net in base 27, using

base change [i] based on digital (9, 36, 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]

(48−27, 48, 32604)-Net in Base 27 — Upper bound on s

There is no (21, 48, 32605)-net in base 27, because

1 times m-reduction [i] would yield (21, 47, 32605)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 18 801531 634140 131222 622860 583433 040438 861794 696357 050421 739483 352579 > 27⁴⁷ [i]