Best Known (5, 5+31, s)-Nets in Base 7
(5, 5+31, 13)-Net over F7 — Constructive and digital
Digital (5, 36, 13)-net over F7, using
- net from sequence [i] based on digital (5, 12)-sequence over F7, using
(5, 5+31, 24)-Net over F7 — Digital
Digital (5, 36, 24)-net over F7, using
- t-expansion [i] based on digital (4, 36, 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
(5, 5+31, 44)-Net over F7 — Upper bound on s (digital)
There is no digital (5, 36, 45)-net over F7, because
- 1 times m-reduction [i] would yield digital (5, 35, 45)-net over F7, but
- 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
- extracting embedded orthogonal array [i] would yield linear OA(735, 45, F7, 30) (dual of [45, 10, 31]-code), but
(5, 5+31, 66)-Net in Base 7 — Upper bound on s
There is no (5, 36, 67)-net in base 7, because
- extracting embedded orthogonal array [i] would yield OA(736, 67, S7, 31), but
- the linear programming bound shows that M ≥ 12334 084449 267931 724496 266386 056538 815394 371109 802439 937447 / 4376 228410 830517 403348 299507 > 736 [i]