Best Known (87−81, 87, s)-Nets in Base 16
(87−81, 87, 65)-Net over F16 — Constructive and digital
Digital (6, 87, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
(87−81, 87, 174)-Net over F16 — Upper bound on s (digital)
There is no digital (6, 87, 175)-net over F16, because
- 1 times m-reduction [i] would yield digital (6, 86, 175)-net over F16, but
- extracting embedded orthogonal array [i] would yield linear OA(1686, 175, F16, 80) (dual of [175, 89, 81]-code), but
- construction Y1 [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
- OA(1689, 175, S16, 83), but
- discarding factors would yield OA(1689, 165, S16, 83), but
- the linear programming bound shows that M ≥ 15352 684767 685611 086882 809761 984160 188561 571509 004484 738285 092713 831920 008077 971129 525920 055364 921813 182353 415776 911815 913281 106245 860502 560348 270257 729557 233664 / 103836 312412 686023 017550 716368 186316 554556 371946 618739 > 1689 [i]
- discarding factors would yield OA(1689, 165, S16, 83), but
- linear OA(1685, 92, F16, 80) (dual of [92, 7, 81]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(1686, 175, F16, 80) (dual of [175, 89, 81]-code), but
(87−81, 87, 176)-Net in Base 16 — Upper bound on s
There is no (6, 87, 177)-net in base 16, because
- extracting embedded orthogonal array [i] would yield OA(1687, 177, S16, 81), but
- the linear programming bound shows that M ≥ 1979 490265 767149 060480 289641 042414 409901 982668 709551 972013 109722 618112 596178 616945 021767 047036 693222 515191 448031 188325 734085 173872 061467 835572 927146 914505 293824 / 3 430222 078227 205946 190279 817987 259978 700370 182828 419579 > 1687 [i]