MinT - Best Known (80−57, 80, s)-Nets in Base 4

Best Known (80−57, 80, s)-Nets in Base 4

(80−57, 80, 34)-Net over F₄ — Constructive and digital

Digital (23, 80, 34)-net over F₄, using

t-expansion [i] based on digital (21, 80, 34)-net over F₄, using
- net from sequence [i] based on digital (21, 33)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
    - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(80−57, 80, 45)-Net over F₄ — Digital

Digital (23, 80, 45)-net over F₄, using

net from sequence [i] based on digital (23, 44)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 23 and N(F) ≥ 45, using
  - a curve from the manYPoints database [i]

(80−57, 80, 115)-Net in Base 4 — Upper bound on s

There is no (23, 80, 116)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁸⁰, 116, S₄, 57), but
- the linear programming bound shows that MÂ â‰¥ 12091 619329 685528 759302 180099 378064 352582 445606 803952 486372 803152 187648 018420 334592 / 8171 190456 251081 322546 110368 265525 > 4⁸⁰ [i]