Best Known (2, 2+14, s)-Nets in Base 5
(2, 2+14, 12)-Net over F5 — Constructive and digital
Digital (2, 16, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
(2, 2+14, 16)-Net over F5 — Upper bound on s (digital)
There is no digital (2, 16, 17)-net over F5, because
- 4 times m-reduction [i] would yield digital (2, 12, 17)-net over F5, but
- extracting embedded orthogonal array [i] would yield linear OA(512, 17, F5, 10) (dual of [17, 5, 11]-code), but
- construction Y1 [i] would yield
- OA(511, 13, S5, 10), but
- the (dual) Plotkin bound shows that M ≥ 732 421875 / 11 > 511 [i]
- linear OA(55, 17, F5, 4) (dual of [17, 12, 5]-code), but
- discarding factors / shortening the dual code would yield linear OA(55, 13, F5, 4) (dual of [13, 8, 5]-code), but
- “Bou†bound on codes from Brouwer’s database [i]
- discarding factors / shortening the dual code would yield linear OA(55, 13, F5, 4) (dual of [13, 8, 5]-code), but
- OA(511, 13, S5, 10), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(512, 17, F5, 10) (dual of [17, 5, 11]-code), but
(2, 2+14, 17)-Net in Base 5 — Upper bound on s
There is no (2, 16, 18)-net in base 5, because
- extracting embedded OOA [i] would yield OOA(516, 18, S5, 2, 14), but
- the linear programming bound for OOAs shows that M ≥ 267 181396 484375 / 1447 > 516 [i]