MinT - Best Known (18, s)-Sequences in Base 4

Best Known (18, s)-Sequences in Base 4

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

Digital (18, 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]

(18, 40)-Sequence over F₄ — Digital

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]

(18, 65)-Sequence in Base 4 — Upper bound on s

There is no (18, 66)-sequence in base 4, because

net from sequence [i] would yield (18, m, 67)-net in base 4 for arbitrarily large m, but
- 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]