MinT - Best Known (65, 65+45, s)-Nets in Base 27

Best Known (65, 65+45, s)-Nets in Base 27

(65, 65+45, 272)-Net over F₂₇ — Constructive and digital

Digital (65, 110, 272)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (8, 23, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]
2. digital (10, 32, 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, 55, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(65, 65+45, 730)-Net in Base 27 — Constructive

(65, 110, 730)-net in base 27, using

27² times duplication [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]

(65, 65+45, 2536)-Net over F₂₇ — Digital

Digital (65, 110, 2536)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+45, 4301652)-Net in Base 27 — Upper bound on s

There is no (65, 110, 4301653)-net in base 27, because

1 times m-reduction [i] would yield (65, 109, 4301653)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 043879 815942 994644 706169 702996 595764 125289 046775 756367 254949 568017 293897 411014 705415 277728 004968 320965 425264 519521 652242 564981 568780 655263 391060 880555 150353 > 27¹⁰⁹ [i]