MinT - Best Known (77−54, 77, s)-Nets in Base 4

Best Known (77−54, 77, s)-Nets in Base 4

(77−54, 77, 34)-Net over F₄ — Constructive and digital

Digital (23, 77, 34)-net over F₄, using

t-expansion [i] based on digital (21, 77, 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]

(77−54, 77, 45)-Net over F₄ — Digital

Digital (23, 77, 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]

(77−54, 77, 131)-Net in Base 4 — Upper bound on s

There is no (23, 77, 132)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁷⁷, 132, S₄, 54), but
- the linear programming bound shows that MÂ â‰¥ 4576 072705 436046 349225 848502 111132 919481 688080 593500 743722 991483 179296 178577 318529 881670 237208 277035 670445 859451 402931 142656 / 184533 840265 255194 708356 812842 195768 632162 832615 941865 395005 248773 469875 098125 > 4⁷⁷ [i]