MinT - Best Known (64, 64+43, s)-Nets in Base 27

Best Known (64, 64+43, s)-Nets in Base 27

(64, 64+43, 276)-Net over F₂₇ — Constructive and digital

Digital (64, 107, 276)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (9, 23, 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]
2. digital (10, 31, 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]
3. digital (10, 53, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(64, 64+43, 730)-Net in Base 27 — Constructive

(64, 107, 730)-net in base 27, using

t-expansion [i] based on (63, 107, 730)-net in base 27, using
- 1 times m-reduction [i] based on (63, 108, 730)-net in base 27, using
  - base change [i] based on digital (36, 81, 730)-net over F₈₁, using
    - net from sequence [i] based on digital (36, 729)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 36 and N(F) ≥ 730, using
        the Hermitian function field over F₈₁ [i]

(64, 64+43, 2835)-Net over F₂₇ — Digital

Digital (64, 107, 2835)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 64+43, 5603990)-Net in Base 27 — Upper bound on s

There is no (64, 107, 5603991)-net in base 27, because

1 times m-reduction [i] would yield (64, 106, 5603991)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 53 034685 350551 023416 149995 067722 858365 487935 035152 700186 292092 711377 359509 079715 063743 944931 637738 581625 271528 458153 106217 873757 200979 463885 328341 639439 > 27¹⁰⁶ [i]