Best Known (27−23, 27, s)-Nets in Base 7
(27−23, 27, 12)-Net over F7 — Constructive and digital
Digital (4, 27, 12)-net over F7, using
- net from sequence [i] based on digital (4, 11)-sequence over F7, using
(27−23, 27, 24)-Net over F7 — Digital
Digital (4, 27, 24)-net over F7, using
- net from sequence [i] based on digital (4, 23)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 4 and N(F) ≥ 24, using
(27−23, 27, 52)-Net over F7 — Upper bound on s (digital)
There is no digital (4, 27, 53)-net over F7, because
- 2 times m-reduction [i] would yield digital (4, 25, 53)-net over F7, but
- extracting embedded orthogonal array [i] would yield linear OA(725, 53, F7, 21) (dual of [53, 28, 22]-code), but
- construction Y1 [i] would yield
- linear OA(724, 29, F7, 21) (dual of [29, 5, 22]-code), but
- construction Y1 [i] would yield
- OA(723, 25, S7, 21), but
- the (dual) Plotkin bound shows that M ≥ 383 162462 761132 828802 / 11 > 723 [i]
- OA(75, 29, S7, 4), but
- the linear programming bound shows that M ≥ 3 384381 / 197 > 75 [i]
- OA(723, 25, S7, 21), but
- construction Y1 [i] would yield
- linear OA(728, 53, F7, 24) (dual of [53, 25, 25]-code), but
- discarding factors / shortening the dual code would yield linear OA(728, 40, F7, 24) (dual of [40, 12, 25]-code), but
- construction Y1 [i] would yield
- linear OA(727, 30, F7, 24) (dual of [30, 3, 25]-code), but
- “Hi4†bound on codes from Brouwer’s database [i]
- linear OA(712, 40, F7, 10) (dual of [40, 28, 11]-code), but
- discarding factors / shortening the dual code would yield linear OA(712, 36, F7, 10) (dual of [36, 24, 11]-code), but
- construction Y1 [i] would yield
- linear OA(711, 15, F7, 10) (dual of [15, 4, 11]-code), but
- “BPM†bound on codes from Brouwer’s database [i]
- linear OA(724, 36, F7, 21) (dual of [36, 12, 22]-code), but
- discarding factors / shortening the dual code would yield linear OA(724, 31, F7, 21) (dual of [31, 7, 22]-code), but
- residual code [i] would yield OA(73, 9, S7, 3), but
- discarding factors / shortening the dual code would yield linear OA(724, 31, F7, 21) (dual of [31, 7, 22]-code), but
- linear OA(711, 15, F7, 10) (dual of [15, 4, 11]-code), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(712, 36, F7, 10) (dual of [36, 24, 11]-code), but
- linear OA(727, 30, F7, 24) (dual of [30, 3, 25]-code), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(728, 40, F7, 24) (dual of [40, 12, 25]-code), but
- linear OA(724, 29, F7, 21) (dual of [29, 5, 22]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(725, 53, F7, 21) (dual of [53, 28, 22]-code), but
(27−23, 27, 68)-Net in Base 7 — Upper bound on s
There is no (4, 27, 69)-net in base 7, because
- extracting embedded orthogonal array [i] would yield OA(727, 69, S7, 23), but
- the linear programming bound shows that M ≥ 691 947938 485240 081566 820608 940481 234739 / 10298 395243 799629 > 727 [i]