MinT - Best Known (19, s)-Sequences in Base 4

Best Known (19, s)-Sequences in Base 4

(19, 32)-Sequence over F₄ — Constructive and digital

Digital (19, 32)-sequence over F₄, using

t-expansion [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]

(19, 40)-Sequence over F₄ — Digital

Digital (19, 40)-sequence over F₄, using

t-expansion [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]

(19, 68)-Sequence in Base 4 — Upper bound on s

There is no (19, 69)-sequence in base 4, because

net from sequence [i] would yield (19, m, 70)-net in base 4 for arbitrarily large m, but
- 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]