MinT - Best Known (122−88, 122, s)-Nets in Base 4

Best Known (122−88, 122, s)-Nets in Base 4

(122−88, 122, 56)-Net over F₄ — Constructive and digital

Digital (34, 122, 56)-net over F₄, using

t-expansion [i] based on digital (33, 122, 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]

(122−88, 122, 65)-Net over F₄ — Digital

Digital (34, 122, 65)-net over F₄, using

t-expansion [i] based on digital (33, 122, 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]

(122−88, 122, 142)-Net in Base 4 — Upper bound on s

There is no (34, 122, 143)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4¹²², 143, S₄, 88), but
- the linear programming bound shows that MÂ â‰¥ 1525 249899 879466 669879 663214 274615 500880 337377 193463 751408 894070 529693 054025 614611 185664 / 47 294304 712863 > 4¹²² [i]