Best Known (4, 33, s)-Nets in Base 8
(4, 33, 25)-Net over F8 — Constructive and digital
Digital (4, 33, 25)-net over F8, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 4 and N(F) ≥ 25, using
(4, 33, 45)-Net over F8 — Upper bound on s (digital)
There is no digital (4, 33, 46)-net over F8, because
- 1 times m-reduction [i] would yield digital (4, 32, 46)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(832, 46, F8, 28) (dual of [46, 14, 29]-code), but
- construction Y1 [i] would yield
- linear OA(831, 34, F8, 28) (dual of [34, 3, 29]-code), but
- “Mas†bound on codes from Brouwer’s database [i]
- linear OA(814, 46, F8, 12) (dual of [46, 32, 13]-code), but
- discarding factors / shortening the dual code would yield linear OA(814, 32, F8, 12) (dual of [32, 18, 13]-code), but
- construction Y1 [i] would yield
- linear OA(813, 16, F8, 12) (dual of [16, 3, 13]-code), but
- “Hi4†bound on codes from Brouwer’s database [i]
- linear OA(818, 32, F8, 16) (dual of [32, 14, 17]-code), but
- discarding factors / shortening the dual code would yield linear OA(818, 27, F8, 16) (dual of [27, 9, 17]-code), but
- residual code [i] would yield OA(82, 10, S8, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 71 > 82 [i]
- residual code [i] would yield OA(82, 10, S8, 2), but
- discarding factors / shortening the dual code would yield linear OA(818, 27, F8, 16) (dual of [27, 9, 17]-code), but
- linear OA(813, 16, F8, 12) (dual of [16, 3, 13]-code), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(814, 32, F8, 12) (dual of [32, 18, 13]-code), but
- linear OA(831, 34, F8, 28) (dual of [34, 3, 29]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(832, 46, F8, 28) (dual of [46, 14, 29]-code), but
(4, 33, 69)-Net in Base 8 — Upper bound on s
There is no (4, 33, 70)-net in base 8, because
- extracting embedded orthogonal array [i] would yield OA(833, 70, S8, 29), but
- the linear programming bound shows that M ≥ 27 702182 052296 607762 357735 488830 788159 349604 220894 445568 / 40 071532 500169 293729 282319 > 833 [i]