MinT - Best Known (251−232, 251, s)-Nets in Base 4

Best Known (251−232, 251, s)-Nets in Base 4

(251−232, 251, 33)-Net over F₄ — Constructive and digital

Digital (19, 251, 33)-net over F₄, using

t-expansion [i] based on digital (15, 251, 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]

(251−232, 251, 41)-Net over F₄ — Digital

Digital (19, 251, 41)-net over F₄, using

t-expansion [i] based on digital (18, 251, 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]

(251−232, 251, 69)-Net in Base 4 — Upper bound on s

There is no (19, 251, 70)-net in base 4, because

45 times m-reduction [i] would yield (19, 206, 70)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²⁰⁶, 70, S₄, 3, 187), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 655767 521039 926610 160299 985568 508871 509777 445500 566969 530371 604912 210544 906420 855058 587696 756197 562568 810482 328823 171437 821952 / 47 > 4²⁰⁶ [i]