Best Known (31−30, 31, s)-Nets in Base 27
(31−30, 31, 38)-Net over F27 — Constructive and digital
Digital (1, 31, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
(31−30, 31, 80)-Net over F27 — Upper bound on s (digital)
There is no digital (1, 31, 81)-net over F27, because
- 3 times m-reduction [i] would yield digital (1, 28, 81)-net over F27, but
- extracting embedded orthogonal array [i] would yield linear OA(2728, 81, F27, 27) (dual of [81, 53, 28]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(2753, 81, F27, 52) (dual of [81, 28, 53]-code), but
- discarding factors / shortening the dual code would yield linear OA(2753, 56, F27, 52) (dual of [56, 3, 53]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(2753, 81, F27, 52) (dual of [81, 28, 53]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2728, 81, F27, 27) (dual of [81, 53, 28]-code), but
(31−30, 31, 135)-Net in Base 27 — Upper bound on s
There is no (1, 31, 136)-net in base 27, because
- 22 times m-reduction [i] would yield (1, 9, 136)-net in base 27, but
- extracting embedded orthogonal array [i] would yield OA(279, 136, S27, 8), but
- the linear programming bound shows that M ≥ 35766 310676 101581 380904 / 4611 122641 > 279 [i]
- extracting embedded orthogonal array [i] would yield OA(279, 136, S27, 8), but