Best Known (81−69, 81, s)-Nets in Base 5
(81−69, 81, 33)-Net over F5 — Constructive and digital
Digital (12, 81, 33)-net over F5, using
- net from sequence [i] based on digital (12, 32)-sequence over F5, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F5 with g(F) = 11, N(F) = 32, and 1 place with degree 2 [i] based on function field F/F5 with g(F) = 11 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
(81−69, 81, 67)-Net over F5 — Upper bound on s (digital)
There is no digital (12, 81, 68)-net over F5, because
- 19 times m-reduction [i] would yield digital (12, 62, 68)-net over F5, but
- extracting embedded orthogonal array [i] would yield linear OA(562, 68, F5, 50) (dual of [68, 6, 51]-code), but
- construction Y1 [i] would yield
- linear OA(561, 64, F5, 50) (dual of [64, 3, 51]-code), but
- OA(56, 68, S5, 4), but
- discarding factors would yield OA(56, 45, S5, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 16021 > 56 [i]
- discarding factors would yield OA(56, 45, S5, 4), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(562, 68, F5, 50) (dual of [68, 6, 51]-code), but
(81−69, 81, 69)-Net in Base 5 — Upper bound on s
There is no (12, 81, 70)-net in base 5, because
- 20 times m-reduction [i] would yield (12, 61, 70)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(561, 70, S5, 49), but
- the linear programming bound shows that M ≥ 191361 683443 691532 602315 419353 544712 066650 390625 / 35929 > 561 [i]
- extracting embedded orthogonal array [i] would yield OA(561, 70, S5, 49), but