Best Known (22, 41, s)-Nets in Base 27
(22, 41, 158)-Net over F27 — Constructive and digital
Digital (22, 41, 158)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 15, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- digital (7, 26, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (6, 15, 76)-net over F27, using
(22, 41, 200)-Net in Base 27 — Constructive
(22, 41, 200)-net in base 27, using
- 271 times duplication [i] based on (21, 40, 200)-net in base 27, using
- base change [i] based on digital (11, 30, 200)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 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 (1, 20, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81 (see above)
- digital (1, 10, 100)-net over F81, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (11, 30, 200)-net over F81, using
(22, 41, 636)-Net over F27 — Digital
Digital (22, 41, 636)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2741, 636, F27, 19) (dual of [636, 595, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2741, 743, F27, 19) (dual of [743, 702, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(2737, 729, F27, 19) (dual of [729, 692, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2727, 729, F27, 14) (dual of [729, 702, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(274, 14, F27, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,27)), using
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- Reed–Solomon code RS(23,27) [i]
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(2741, 743, F27, 19) (dual of [743, 702, 20]-code), using
(22, 41, 366766)-Net in Base 27 — Upper bound on s
There is no (22, 41, 366767)-net in base 27, because
- 1 times m-reduction [i] would yield (22, 40, 366767)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 1797 031509 107956 572800 819911 285760 560017 570523 980380 364535 > 2740 [i]