MinT - Best Known (122−89, 122, s)-Nets in Base 4

Best Known (122−89, 122, s)-Nets in Base 4

(122−89, 122, 56)-Net over F₄ — Constructive and digital

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]

(122−89, 122, 65)-Net over F₄ — Digital

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]

(122−89, 122, 138)-Net in Base 4 — Upper bound on s

There is no (33, 122, 139)-net in base 4, because

1 times m-reduction [i] would yield (33, 121, 139)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹²¹, 139, S₄, 88), but
  - the linear programming bound shows that MÂ â‰¥ 89 823087 987600 850167 052548 419117 527544 927341 267278 328150 741318 211793 982911 705155 371008 / 8 583431 601745 > 4¹²¹ [i]