MinT - Best Known (226−208, 226, s)-Nets in Base 4

Best Known (226−208, 226, s)-Nets in Base 4

(226−208, 226, 33)-Net over F₄ — Constructive and digital

Digital (18, 226, 33)-net over F₄, using

t-expansion [i] based on digital (15, 226, 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]

(226−208, 226, 41)-Net over F₄ — Digital

Digital (18, 226, 41)-net over F₄, using

net from sequence [i] based on digital (18, 40)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 18 and N(F) ≥ 41, using
  - Drinfeld modules of rank 1 [i]

(226−208, 226, 66)-Net in Base 4 — Upper bound on s

There is no (18, 226, 67)-net in base 4, because

29 times m-reduction [i] would yield (18, 197, 67)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁹⁷, 67, S₄, 3, 179), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 282433 580415 755626 993616 159437 829423 754811 571090 696321 715853 367121 633333 658092 005662 617460 515015 075313 301909 434519 257088 / 5 > 4¹⁹⁷ [i]