MinT - Best Known (4, s)-Sequences in Base 16

Best Known (4, s)-Sequences in Base 16

(4, 44)-Sequence over F₁₆ — Constructive and digital

Digital (4, 44)-sequence over F₁₆, using

Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
- a function field by GarcÃa and Quoos [i]

(4, 75)-Sequence over F₁₆ — Upper bound on s (digital)

There is no digital (4, 76)-sequence over F₁₆, because

net from sequence [i] would yield digital (4, m, 77)-net over F₁₆ for arbitrarily large m, but
- m-reduction [i] would yield digital (4, 68, 77)-net over F₁₆, but
  - extracting embedded orthogonal array [i] would yield linear OA(16⁶⁸, 77, F₁₆, 64) (dual of [77, 9, 65]-code), but
    - constructionÂ Y1 [i] would yield
      1. OA(16⁶⁷, 69, S₁₆, 64), but
        the (dual) Plotkin bound shows that MÂ â‰¥ 7588 550360 256754 183279 148073 529370 729071 901715 047420 004889 892225 542594 864082 845696 / 13 > 16⁶⁷ [i]
      2. OA(16⁹, 77, S₁₆, 8), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 68757 087631 > 16⁹ [i]

(4, 79)-Sequence in Base 16 — Upper bound on s

There is no (4, 80)-sequence in base 16, because

net from sequence [i] would yield (4, m, 81)-net in base 16 for arbitrarily large m, but
- m-reduction [i] would yield (4, 77, 81)-net in base 16, but
  - extracting embedded orthogonal array [i] would yield OA(16⁷⁷, 81, S₁₆, 73), but
    - the linear programming bound shows that MÂ â‰¥ 20 058253 259194 796242 490750 709498 450326 191022 155255 442417 783937 598455 532112 408827 665059 378347 638784 / 37999 > 16⁷⁷ [i]