Best Known (69−67, 69, s)-Nets in Base 27
(69−67, 69, 48)-Net over F27 — Constructive and digital
Digital (2, 69, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
(69−67, 69, 83)-Net over F27 — Upper bound on s (digital)
There is no digital (2, 69, 84)-net over F27, because
- 13 times m-reduction [i] would yield digital (2, 56, 84)-net over F27, but
- extracting embedded orthogonal array [i] would yield linear OA(2756, 84, F27, 54) (dual of [84, 28, 55]-code), but
- residual code [i] would yield OA(272, 29, S27, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 755 > 272 [i]
- residual code [i] would yield OA(272, 29, S27, 2), but
- extracting embedded orthogonal array [i] would yield linear OA(2756, 84, F27, 54) (dual of [84, 28, 55]-code), but
(69−67, 69, 146)-Net in Base 27 — Upper bound on s
There is no (2, 69, 147)-net in base 27, because
- extracting embedded orthogonal array [i] would yield OA(2769, 147, S27, 67), but
- the linear programming bound shows that M ≥ 127789 917045 494956 472582 651423 262858 314293 535306 210523 849364 690935 169892 385416 786912 871478 257824 513218 384227 / 218 192297 > 2769 [i]