Best Known (3, 89, s)-Nets in Base 32
(3, 89, 64)-Net over F32 — Constructive and digital
Digital (3, 89, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
(3, 89, 186)-Net over F32 — Upper bound on s (digital)
There is no digital (3, 89, 187)-net over F32, because
- 1 times m-reduction [i] would yield digital (3, 88, 187)-net over F32, but
- extracting embedded orthogonal array [i] would yield linear OA(3288, 187, F32, 85) (dual of [187, 99, 86]-code), but
- construction Y1 [i] would yield
- OA(3287, 91, S32, 85), but
- the linear programming bound shows that M ≥ 3218 248804 644806 290772 103657 191613 826913 486259 137011 802066 676331 137713 410455 220883 164990 453062 699224 855954 368127 744971 436177 620882 948096 / 34443 > 3287 [i]
- linear OA(3299, 187, F32, 96) (dual of [187, 88, 97]-code), but
- discarding factors / shortening the dual code would yield linear OA(3299, 132, F32, 96) (dual of [132, 33, 97]-code), but
- residual code [i] would yield OA(323, 35, S32, 3), but
- 1 times truncation [i] would yield OA(322, 34, S32, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 1055 > 322 [i]
- 1 times truncation [i] would yield OA(322, 34, S32, 2), but
- residual code [i] would yield OA(323, 35, S32, 3), but
- discarding factors / shortening the dual code would yield linear OA(3299, 132, F32, 96) (dual of [132, 33, 97]-code), but
- OA(3287, 91, S32, 85), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(3288, 187, F32, 85) (dual of [187, 99, 86]-code), but
(3, 89, 298)-Net in Base 32 — Upper bound on s
There is no (3, 89, 299)-net in base 32, because
- extracting embedded orthogonal array [i] would yield OA(3289, 299, S32, 86), but
- the linear programming bound shows that M ≥ 2 360435 264521 435753 863984 728525 460417 636601 778557 757701 268173 418031 085326 973077 668398 961432 098396 263745 738727 715106 031793 750019 589380 141259 838614 726353 354752 / 25927 420448 859780 644999 > 3289 [i]