Best Known (33−26, 33, s)-Nets in Base 5
(33−26, 33, 22)-Net over F5 — Constructive and digital
Digital (7, 33, 22)-net over F5, using
- net from sequence [i] based on digital (7, 21)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 7 and N(F) ≥ 22, using
(33−26, 33, 65)-Net over F5 — Upper bound on s (digital)
There is no digital (7, 33, 66)-net over F5, because
- 1 times m-reduction [i] would yield digital (7, 32, 66)-net over F5, but
- extracting embedded orthogonal array [i] would yield linear OA(532, 66, F5, 25) (dual of [66, 34, 26]-code), but
- construction Y1 [i] would yield
- linear OA(531, 39, F5, 25) (dual of [39, 8, 26]-code), but
- construction Y1 [i] would yield
- linear OA(530, 33, F5, 25) (dual of [33, 3, 26]-code), but
- OA(58, 39, S5, 6), but
- discarding factors would yield OA(58, 34, S5, 6), but
- the Rao or (dual) Hamming bound shows that M ≥ 392089 > 58 [i]
- discarding factors would yield OA(58, 34, S5, 6), but
- construction Y1 [i] would yield
- OA(534, 66, S5, 27), but
- discarding factors would yield OA(534, 65, S5, 27), but
- the linear programming bound shows that M ≥ 64 241548 253700 687835 401117 197195 381929 859807 132743 299007 415771 484375 / 105 626721 073484 056311 392841 841080 747283 817551 > 534 [i]
- discarding factors would yield OA(534, 65, S5, 27), but
- linear OA(531, 39, F5, 25) (dual of [39, 8, 26]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(532, 66, F5, 25) (dual of [66, 34, 26]-code), but
(33−26, 33, 68)-Net in Base 5 — Upper bound on s
There is no (7, 33, 69)-net in base 5, because
- extracting embedded orthogonal array [i] would yield OA(533, 69, S5, 26), but
- the linear programming bound shows that M ≥ 220944 006465 288816 531507 328552 487121 062961 679566 302336 752414 703369 140625 / 1 789706 658944 937570 975682 316278 255197 219983 298407 > 533 [i]