Best Known (35−31, 35, s)-Nets in Base 7
(35−31, 35, 12)-Net over F7 — Constructive and digital
Digital (4, 35, 12)-net over F7, using
- net from sequence [i] based on digital (4, 11)-sequence over F7, using
(35−31, 35, 24)-Net over F7 — Digital
Digital (4, 35, 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
(35−31, 35, 36)-Net over F7 — Upper bound on s (digital)
There is no digital (4, 35, 37)-net over F7, because
- 3 times m-reduction [i] would yield digital (4, 32, 37)-net over F7, but
- extracting embedded orthogonal array [i] would yield linear OA(732, 37, F7, 28) (dual of [37, 5, 29]-code), but
- construction Y1 [i] would yield
- OA(731, 33, S7, 28), but
- the (dual) Plotkin bound shows that M ≥ 5522 138371 219603 231526 496005 / 29 > 731 [i]
- OA(75, 37, S7, 4), but
- discarding factors would yield OA(75, 31, S7, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 16927 > 75 [i]
- discarding factors would yield OA(75, 31, S7, 4), but
- OA(731, 33, S7, 28), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(732, 37, F7, 28) (dual of [37, 5, 29]-code), but
(35−31, 35, 39)-Net in Base 7 — Upper bound on s
There is no (4, 35, 40)-net in base 7, because
- extracting embedded orthogonal array [i] would yield OA(735, 40, S7, 31), but
- the linear programming bound shows that M ≥ 18 562115 921017 574302 453163 671207 / 37 > 735 [i]