Best Known (110−106, 110, s)-Nets in Base 7
(110−106, 110, 12)-Net over F7 — Constructive and digital
Digital (4, 110, 12)-net over F7, using
- net from sequence [i] based on digital (4, 11)-sequence over F7, using
(110−106, 110, 24)-Net over F7 — Digital
Digital (4, 110, 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
(110−106, 110, 36)-Net over F7 — Upper bound on s (digital)
There is no digital (4, 110, 37)-net over F7, because
- 78 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
(110−106, 110, 38)-Net in Base 7 — Upper bound on s
There is no (4, 110, 39)-net in base 7, because
- 36 times m-reduction [i] would yield (4, 74, 39)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(774, 39, S7, 2, 70), but
- the LP bound with quadratic polynomials shows that M ≥ 30665 141124 411175 854056 235783 580684 649065 396053 855752 734074 479561 / 71 > 774 [i]
- extracting embedded OOA [i] would yield OOA(774, 39, S7, 2, 70), but