MinT - Best Known (5, s)-Sequences in Base 16

Best Known (5, s)-Sequences in Base 16

(5, 48)-Sequence over F₁₆ — Constructive and digital

Digital (5, 48)-sequence over F₁₆, using

Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
- a function field by SÃ©mirat [i]

(5, 90)-Sequence over F₁₆ — Upper bound on s (digital)

There is no digital (5, 91)-sequence over F₁₆, because

net from sequence [i] would yield digital (5, m, 92)-net over F₁₆ for arbitrarily large m, but
- m-reduction [i] would yield digital (5, 85, 92)-net over F₁₆, but
  - extracting embedded orthogonal array [i] would yield linear OA(16⁸⁵, 92, F₁₆, 80) (dual of [92, 7, 81]-code), but
    - constructionÂ Y1 [i] would yield
      1. OA(16⁸⁴, 86, S₁₆, 80), but
        the (dual) Plotkin bound shows that MÂ â‰¥ 4 479489 484355 608421 114884 561136 888556 243290 994469 299069 799978 201927 583742 360321 890761 754986 543214 231552 / 27 > 16⁸⁴ [i]
      2. OA(16⁷, 92, S₁₆, 6), but
        discarding factors would yield OA(16⁷, 80, S₁₆, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 278 002201 > 16⁷ [i]

(5, 92)-Sequence in Base 16 — Upper bound on s

There is no (5, 93)-sequence in base 16, because

net from sequence [i] would yield (5, m, 94)-net in base 16 for arbitrarily large m, but
- m-reduction [i] would yield (5, 87, 94)-net in base 16, but
  - extracting embedded orthogonal array [i] would yield OA(16⁸⁷, 94, S₁₆, 82), but
    - the linear programming bound shows that MÂ â‰¥ 1423 730548 850924 712119 626065 534885 906064 402055 196015 536620 335548 648541 339943 696537 336009 277177 066502 191140 831232 / 2 409407 > 16⁸⁷ [i]