Best Known (35−30, 35, s)-Nets in Base 7
(35−30, 35, 13)-Net over F7 — Constructive and digital
Digital (5, 35, 13)-net over F7, using
- net from sequence [i] based on digital (5, 12)-sequence over F7, using
(35−30, 35, 24)-Net over F7 — Digital
Digital (5, 35, 24)-net over F7, using
- t-expansion [i] based on 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
- net from sequence [i] based on digital (4, 23)-sequence over F7, using
(35−30, 35, 44)-Net over F7 — Upper bound on s (digital)
There is no digital (5, 35, 45)-net over F7, because
- extracting embedded orthogonal array [i] would yield linear OA(735, 45, F7, 30) (dual of [45, 10, 31]-code), but
- construction Y1 [i] would yield
- linear OA(734, 37, F7, 30) (dual of [37, 3, 31]-code), but
- “Hi4†bound on codes from Brouwer’s database [i]
- linear OA(710, 45, F7, 8) (dual of [45, 35, 9]-code), but
- discarding factors / shortening the dual code would yield linear OA(710, 42, F7, 8) (dual of [42, 32, 9]-code), but
- construction Y1 [i] would yield
- linear OA(79, 14, F7, 8) (dual of [14, 5, 9]-code), but
- “MPa†bound on codes from Brouwer’s database [i]
- linear OA(732, 42, F7, 28) (dual of [42, 10, 29]-code), but
- discarding factors / shortening the dual code would yield linear OA(732, 38, F7, 28) (dual of [38, 6, 29]-code), but
- linear OA(79, 14, F7, 8) (dual of [14, 5, 9]-code), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(710, 42, F7, 8) (dual of [42, 32, 9]-code), but
- linear OA(734, 37, F7, 30) (dual of [37, 3, 31]-code), but
- construction Y1 [i] would yield
(35−30, 35, 71)-Net in Base 7 — Upper bound on s
There is no (5, 35, 72)-net in base 7, because
- extracting embedded orthogonal array [i] would yield OA(735, 72, S7, 30), but
- the linear programming bound shows that M ≥ 486730 269193 215173 891992 037638 825119 564889 847855 894659 664752 / 1 244240 452248 670658 546570 605097 > 735 [i]