MinT - Best Known (219−202, 219, s)-Nets in Base 4

Best Known (219−202, 219, s)-Nets in Base 4

(219−202, 219, 33)-Net over F₄ — Constructive and digital

Digital (17, 219, 33)-net over F₄, using

t-expansion [i] based on digital (15, 219, 33)-net over F₄, using
- net from sequence [i] based on digital (15, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(219−202, 219, 40)-Net over F₄ — Digital

Digital (17, 219, 40)-net over F₄, using

net from sequence [i] based on digital (17, 39)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 17 and N(F) ≥ 40, using
  - Drinfeld modules of rank 1 [i]

(219−202, 219, 62)-Net in Base 4 — Upper bound on s

There is no (17, 219, 63)-net in base 4, because

34 times m-reduction [i] would yield (17, 185, 63)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁸⁵, 63, S₄, 3, 168), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 413644 108018 789698 761718 438551 312351 957624 215974 124562 187933 744234 024911 175385 755930 374377 187512 542127 608643 452928 / 169 > 4¹⁸⁵ [i]