MinT - Best Known (61, 61+46, s)-Nets in Base 27

Best Known (61, 61+46, s)-Nets in Base 27

(61, 61+46, 252)-Net over F₂₇ — Constructive and digital

Digital (61, 107, 252)-net over F₂₇, using

1 times m-reduction [i] based on digital (61, 108, 252)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (7, 22, 82)-net over F₂₇, using
    - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
        a function field by SÃ©mirat [i]
  2. digital (7, 30, 82)-net over F₂₇, using
    - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
  3. digital (9, 56, 88)-net over F₂₇, using
    - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
        a function field by SÃ©mirat [i]

(61, 61+46, 408)-Net in Base 27 — Constructive

(61, 107, 408)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (1, 24, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. (37, 83, 370)-net in base 27, using
  - 1 times m-reduction [i] based on (37, 84, 370)-net in base 27, using
    - base change [i] based on digital (16, 63, 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]

(61, 61+46, 1739)-Net over F₂₇ — Digital

Digital (61, 107, 1739)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(61, 61+46, 1653609)-Net in Base 27 — Upper bound on s

There is no (61, 107, 1653610)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1431 942020 930595 645019 476894 352365 031503 825541 576262 251121 430909 981715 689451 609090 044098 709095 316823 163090 035829 219031 082927 374310 296597 105585 388360 456473 > 27¹⁰⁷ [i]