Best Known (91−85, 91, s)-Nets in Base 16
(91−85, 91, 65)-Net over F16 — Constructive and digital
Digital (6, 91, 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
(91−85, 91, 149)-Net over F16 — Upper bound on s (digital)
There is no digital (6, 91, 150)-net over F16, because
- 1 times m-reduction [i] would yield digital (6, 90, 150)-net over F16, but
- extracting embedded orthogonal array [i] would yield linear OA(1690, 150, F16, 84) (dual of [150, 60, 85]-code), but
- construction Y1 [i] would yield
- linear OA(1689, 94, F16, 84) (dual of [94, 5, 85]-code), but
- construction Y1 [i] would yield
- OA(1688, 90, S16, 84), but
- 4 times truncation [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]
- 4 times truncation [i] would yield OA(1684, 86, S16, 80), but
- OA(165, 94, S16, 4), but
- the linear programming bound shows that M ≥ 2179 760128 / 2071 > 165 [i]
- OA(1688, 90, S16, 84), but
- construction Y1 [i] would yield
- OA(1660, 150, S16, 56), but
- discarding factors would yield OA(1660, 147, S16, 56), but
- the linear programming bound shows that M ≥ 939765 966922 797477 814596 181274 291363 410391 878783 132568 319700 182987 088048 908156 194099 962624 995855 466829 447168 / 521248 355323 503233 415469 012835 684063 > 1660 [i]
- discarding factors would yield OA(1660, 147, S16, 56), but
- linear OA(1689, 94, F16, 84) (dual of [94, 5, 85]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(1690, 150, F16, 84) (dual of [150, 60, 85]-code), but
(91−85, 91, 151)-Net in Base 16 — Upper bound on s
There is no (6, 91, 152)-net in base 16, because
- extracting embedded orthogonal array [i] would yield OA(1691, 152, S16, 85), but
- the linear programming bound shows that M ≥ 2 278206 633061 117229 098518 919813 179843 089019 560647 409807 403603 381299 543561 427491 281877 305424 620815 339341 203582 514016 289427 818186 181942 733172 772853 252096 / 60571 801747 214090 732136 958267 349615 729255 > 1691 [i]