Best Known (5, 5+∞, s)-Nets in Base 16
(5, 5+∞, 49)-Net over F16 — Constructive and digital
Digital (5, m, 49)-net over F16 for arbitrarily large m, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
(5, 5+∞, 91)-Net over F16 — Upper bound on s (digital)
There is no digital (5, m, 92)-net over F16 for arbitrarily large m, because
- 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
- construction Y1 [i] would yield
- OA(1684, 86, S16, 80), but
- the (dual) Plotkin bound shows that M ≥ 4 479489 484355 608421 114884 561136 888556 243290 994469 299069 799978 201927 583742 360321 890761 754986 543214 231552 / 27 > 1684 [i]
- OA(167, 92, S16, 6), but
- discarding factors would yield OA(167, 80, S16, 6), but
- the Rao or (dual) Hamming bound shows that M ≥ 278 002201 > 167 [i]
- discarding factors would yield OA(167, 80, S16, 6), but
- OA(1684, 86, S16, 80), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(1685, 92, F16, 80) (dual of [92, 7, 81]-code), but
(5, 5+∞, 93)-Net in Base 16 — Upper bound on s
There is no (5, m, 94)-net in base 16 for arbitrarily large m, because
- 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 M ≥ 1423 730548 850924 712119 626065 534885 906064 402055 196015 536620 335548 648541 339943 696537 336009 277177 066502 191140 831232 / 2 409407 > 1687 [i]
- extracting embedded orthogonal array [i] would yield OA(1687, 94, S16, 82), but