MinT - Best Known (48, 48+53, s)-Nets in Base 27

Best Known (48, 48+53, s)-Nets in Base 27

(48, 48+53, 192)-Net over F₂₇ — Constructive and digital

Digital (48, 101, 192)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (11, 37, 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 (11, 64, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(48, 48+53, 370)-Net in Base 27 — Constructive

(48, 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]

(48, 48+53, 468)-Net over F₂₇ — Digital

Digital (48, 101, 468)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(48, 48+53, 129873)-Net in Base 27 — Upper bound on s

There is no (48, 101, 129874)-net in base 27, because

1 times m-reduction [i] would yield (48, 100, 129874)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 136911 300731 403652 574825 647880 183071 009201 604975 170758 464853 707952 940230 832277 694508 996024 982050 912111 733162 746351 838105 415801 736385 147892 668325 > 27¹⁰⁰ [i]