Best Known (28−23, 28, s)-Nets in Base 5
(28−23, 28, 20)-Net over F5 — Constructive and digital
Digital (5, 28, 20)-net over F5, using
- net from sequence [i] based on digital (5, 19)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 5 and N(F) ≥ 20, using
(28−23, 28, 33)-Net over F5 — Upper bound on s (digital)
There is no digital (5, 28, 34)-net over F5, because
- 3 times m-reduction [i] would yield digital (5, 25, 34)-net over F5, but
- extracting embedded orthogonal array [i] would yield linear OA(525, 34, F5, 20) (dual of [34, 9, 21]-code), but
- residual code [i] would yield linear OA(55, 13, F5, 4) (dual of [13, 8, 5]-code), but
- “Bou†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(55, 13, F5, 4) (dual of [13, 8, 5]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(525, 34, F5, 20) (dual of [34, 9, 21]-code), but
(28−23, 28, 39)-Net in Base 5 — Upper bound on s
There is no (5, 28, 40)-net in base 5, because
- extracting embedded orthogonal array [i] would yield OA(528, 40, S5, 23), but
- the linear programming bound shows that M ≥ 9969 272650 778293 609619 140625 / 261 134328 > 528 [i]