MinT - Best Known (51, 51+45, s)-Nets in Base 27

Best Known (51, 51+45, s)-Nets in Base 27

(51, 51+45, 204)-Net over F₂₇ — Constructive and digital

Digital (51, 96, 204)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (11, 33, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]
2. digital (18, 63, 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]

(51, 51+45, 370)-Net in Base 27 — Constructive

(51, 96, 370)-net in base 27, using

t-expansion [i] based on (43, 96, 370)-net in base 27, using
- 12 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]

(51, 51+45, 903)-Net over F₂₇ — Digital

Digital (51, 96, 903)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+45, 528152)-Net in Base 27 — Upper bound on s

There is no (51, 96, 528153)-net in base 27, because

1 times m-reduction [i] would yield (51, 95, 528153)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 9540 405579 973288 136797 750443 527515 740162 108474 149964 525560 394064 369796 434663 964762 093035 974173 906181 242341 934716 013382 320080 994297 916153 > 27⁹⁵ [i]