MinT - Best Known (4, 123, s)-Nets in Base 16

Best Known (4, 123, s)-Nets in Base 16

(4, 123, 45)-Net over F₁₆ — Constructive and digital

Digital (4, 123, 45)-net over F₁₆, using

net from sequence [i] based on 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, 123, 76)-Net over F₁₆ — Upper bound on s (digital)

There is no digital (4, 123, 77)-net over F₁₆, because

55 times 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, 123, 80)-Net in Base 16 — Upper bound on s

There is no (4, 123, 81)-net in base 16, because

46 times 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]