MinT - Best Known (52, 52+49, s)-Nets in Base 27

Best Known (52, 52+49, s)-Nets in Base 27

(52, 52+49, 202)-Net over F₂₇ — Constructive and digital

Digital (52, 101, 202)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 34, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]
2. digital (18, 67, 108)-net over F₂₇, using
  - net from sequence [i] based on digital (18, 107)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 108, using
      - F₃ from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F₂₇ [i]

(52, 52+49, 370)-Net in Base 27 — Constructive

(52, 101, 370)-net in base 27, using

t-expansion [i] based on (43, 101, 370)-net in base 27, using
- 7 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(52, 52+49, 763)-Net over F₂₇ — Digital

Digital (52, 101, 763)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+49, 347047)-Net in Base 27 — Upper bound on s

There is no (52, 101, 347048)-net in base 27, because

1 times m-reduction [i] would yield (52, 100, 347048)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 136891 626243 824842 797960 910326 496377 014107 807692 257043 878891 354902 477008 238884 139669 171678 054489 491590 605303 681404 513923 041941 786803 123651 525889 > 27¹⁰⁰ [i]