Best Known (22, 22+15, s)-Nets in Base 27
(22, 22+15, 196)-Net over F27 — Constructive and digital
Digital (22, 37, 196)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 28)-net over F27, using
- digital (0, 2, 28)-net over F27 (see above)
- digital (0, 3, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- digital (0, 3, 28)-net over F27 (see above)
- digital (0, 5, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 7, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 15, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
(22, 22+15, 250)-Net in Base 27 — Constructive
(22, 37, 250)-net in base 27, using
- 271 times duplication [i] based on (21, 36, 250)-net in base 27, using
- base change [i] based on digital (12, 27, 250)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (4, 19, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- digital (1, 8, 100)-net over F81, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (12, 27, 250)-net over F81, using
(22, 22+15, 1418)-Net over F27 — Digital
Digital (22, 37, 1418)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2737, 1418, F27, 15) (dual of [1418, 1381, 16]-code), using
- 1368 step Varšamov–Edel lengthening with (ri) = (4, 1, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 8 times 0, 1, 12 times 0, 1, 16 times 0, 1, 20 times 0, 1, 26 times 0, 1, 34 times 0, 1, 43 times 0, 1, 54 times 0, 1, 70 times 0, 1, 90 times 0, 1, 113 times 0, 1, 144 times 0, 1, 183 times 0, 1, 232 times 0, 1, 294 times 0) [i] based on linear OA(2715, 28, F27, 15) (dual of [28, 13, 16]-code or 28-arc in PG(14,27)), using
- extended Reed–Solomon code RSe(13,27) [i]
- the expurgated narrow-sense BCH-code C(I) with length 28 | 272−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 1368 step Varšamov–Edel lengthening with (ri) = (4, 1, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 8 times 0, 1, 12 times 0, 1, 16 times 0, 1, 20 times 0, 1, 26 times 0, 1, 34 times 0, 1, 43 times 0, 1, 54 times 0, 1, 70 times 0, 1, 90 times 0, 1, 113 times 0, 1, 144 times 0, 1, 183 times 0, 1, 232 times 0, 1, 294 times 0) [i] based on linear OA(2715, 28, F27, 15) (dual of [28, 13, 16]-code or 28-arc in PG(14,27)), using
(22, 22+15, 2987061)-Net in Base 27 — Upper bound on s
There is no (22, 37, 2987062)-net in base 27, because
- 1 times m-reduction [i] would yield (22, 36, 2987062)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 3381 394839 550405 297097 676899 467609 316590 091944 131177 > 2736 [i]