Best Known (4, 4+27, s)-Nets in Base 8
(4, 4+27, 25)-Net over F8 — Constructive and digital
Digital (4, 31, 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, 4+27, 62)-Net over F8 — Upper bound on s (digital)
There is no digital (4, 31, 63)-net over F8, because
- extracting embedded orthogonal array [i] would yield linear OA(831, 63, F8, 27) (dual of [63, 32, 28]-code), but
- construction Y1 [i] would yield
- linear OA(830, 35, F8, 27) (dual of [35, 5, 28]-code), but
- construction Y1 [i] would yield
- OA(829, 31, S8, 27), but
- 3 times truncation [i] would yield OA(826, 28, S8, 24), but
- the (dual) Plotkin bound shows that M ≥ 9 671406 556917 033397 649408 / 25 > 826 [i]
- 3 times truncation [i] would yield OA(826, 28, S8, 24), but
- OA(85, 35, S8, 4), but
- the linear programming bound shows that M ≥ 1 306624 / 39 > 85 [i]
- OA(829, 31, S8, 27), but
- construction Y1 [i] would yield
- linear OA(832, 63, F8, 28) (dual of [63, 31, 29]-code), but
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code would yield linear OA(832, 46, F8, 28) (dual of [46, 14, 29]-code), but
- linear OA(830, 35, F8, 27) (dual of [35, 5, 28]-code), but
- construction Y1 [i] would yield
(4, 4+27, 76)-Net in Base 8 — Upper bound on s
There is no (4, 31, 77)-net in base 8, because
- extracting embedded orthogonal array [i] would yield OA(831, 77, S8, 27), but
- the linear programming bound shows that M ≥ 529 927605 055613 384620 673348 236892 370160 844800 / 50712 044326 091649 > 831 [i]