Best Known (28−21, 28, s)-Nets in Base 4
(28−21, 28, 21)-Net over F4 — Constructive and digital
Digital (7, 28, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
(28−21, 28, 40)-Net over F4 — Upper bound on s (digital)
There is no digital (7, 28, 41)-net over F4, because
- 1 times m-reduction [i] would yield digital (7, 27, 41)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(427, 41, F4, 20) (dual of [41, 14, 21]-code), but
- construction Y1 [i] would yield
- linear OA(426, 31, F4, 20) (dual of [31, 5, 21]-code), but
- “Bou†bound on codes from Brouwer’s database [i]
- linear OA(414, 41, F4, 10) (dual of [41, 27, 11]-code), but
- discarding factors / shortening the dual code would yield linear OA(414, 32, F4, 10) (dual of [32, 18, 11]-code), but
- construction Y1 [i] would yield
- linear OA(413, 18, F4, 10) (dual of [18, 5, 11]-code), but
- “Liz†bound on codes from Brouwer’s database [i]
- linear OA(418, 32, F4, 14) (dual of [32, 14, 15]-code), but
- construction Y1 [i] would yield
- linear OA(417, 21, F4, 14) (dual of [21, 4, 15]-code), but
- 2 times truncation [i] would yield linear OA(415, 19, F4, 12) (dual of [19, 4, 13]-code), but
- construction Y1 [i] would yield
- OA(414, 16, S4, 12), but
- the (dual) Plotkin bound shows that M ≥ 4294 967296 / 13 > 414 [i]
- linear OA(44, 19, F4, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,4)), but
- discarding factors / shortening the dual code would yield linear OA(44, 18, F4, 3) (dual of [18, 14, 4]-code or 18-cap in PG(3,4)), but
- OA(414, 16, S4, 12), but
- construction Y1 [i] would yield
- 2 times truncation [i] would yield linear OA(415, 19, F4, 12) (dual of [19, 4, 13]-code), but
- OA(414, 32, S4, 11), but
- the linear programming bound shows that M ≥ 1976 281134 530560 / 7 232221 > 414 [i]
- linear OA(417, 21, F4, 14) (dual of [21, 4, 15]-code), but
- construction Y1 [i] would yield
- linear OA(413, 18, F4, 10) (dual of [18, 5, 11]-code), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(414, 32, F4, 10) (dual of [32, 18, 11]-code), but
- linear OA(426, 31, F4, 20) (dual of [31, 5, 21]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(427, 41, F4, 20) (dual of [41, 14, 21]-code), but
(28−21, 28, 53)-Net in Base 4 — Upper bound on s
There is no (7, 28, 54)-net in base 4, because
- extracting embedded orthogonal array [i] would yield OA(428, 54, S4, 21), but
- the linear programming bound shows that M ≥ 3174 580975 536582 195590 136001 131574 958772 796721 725440 / 41960 362570 094555 942758 209044 973449 > 428 [i]