MinT - Best Known (180−139, 180, s)-Nets in Base 4

Best Known (180−139, 180, s)-Nets in Base 4

(180−139, 180, 56)-Net over F₄ — Constructive and digital

Digital (41, 180, 56)-net over F₄, using

t-expansion [i] based on digital (33, 180, 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]

(180−139, 180, 75)-Net over F₄ — Digital

Digital (41, 180, 75)-net over F₄, using

t-expansion [i] based on digital (40, 180, 75)-net over F₄, using
- net from sequence [i] based on digital (40, 74)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
    - Drinfeld modules of rank 1 [i]

(180−139, 180, 172)-Net over F₄ — Upper bound on s (digital)

There is no digital (41, 180, 173)-net over F₄, because

11 times m-reduction [i] would yield digital (41, 169, 173)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁶⁹, 173, F₄, 128) (dual of [173, 4, 129]-code), but
  - Griesmer bound [i]

(180−139, 180, 178)-Net in Base 4 — Upper bound on s

There is no (41, 180, 179)-net in base 4, because

5 times m-reduction [i] would yield (41, 175, 179)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁷⁵, 179, S₄, 134), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 146783 911423 364576 743092 537299 333564 210980 159306 769991 919205 685720 763064 069663 027716 481187 399048 043939 495936 / 45 > 4¹⁷⁵ [i]