MinT - Best Known (34, 34+109, s)-Nets in Base 4

Best Known (34, 34+109, s)-Nets in Base 4

(34, 34+109, 56)-Net over F₄ — Constructive and digital

Digital (34, 143, 56)-net over F₄, using

t-expansion [i] based on digital (33, 143, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(34, 34+109, 65)-Net over F₄ — Digital

Digital (34, 143, 65)-net over F₄, using

t-expansion [i] based on digital (33, 143, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(34, 34+109, 140)-Net in Base 4 — Upper bound on s

There is no (34, 143, 141)-net in base 4, because

16 times m-reduction [i] would yield (34, 127, 141)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹²⁷, 141, S₄, 93), but
  - the linear programming bound shows that MÂ â‰¥ 146 430564 889104 392909 100261 013841 119930 853683 468983 778269 356582 857126 296146 058611 261440 / 5003 564797 > 4¹²⁷ [i]