MinT - Best Known (78, 78+97, s)-Nets in Base 3

Best Known (78, 78+97, s)-Nets in Base 3

(78, 78+97, 56)-Net over F₃ — Constructive and digital

Digital (78, 175, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 63, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (15, 112, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(78, 78+97, 84)-Net over F₃ — Digital

Digital (78, 175, 84)-net over F₃, using

t-expansion [i] based on digital (71, 175, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(78, 78+97, 456)-Net in Base 3 — Upper bound on s

There is no (78, 175, 457)-net in base 3, because

1 times m-reduction [i] would yield (78, 174, 457)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 104519 719697 958194 313171 749934 006155 528820 151893 023211 981916 526632 847839 944420 578753 > 3¹⁷⁴ [i]