Best Known (20, 20+17, s)-Nets in Base 27
(20, 20+17, 152)-Net over F27 — Constructive and digital
Digital (20, 37, 152)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 14, 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 (6, 23, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27 (see above)
- digital (6, 14, 76)-net over F27, using
(20, 20+17, 200)-Net in Base 27 — Constructive
(20, 37, 200)-net in base 27, using
- 271 times duplication [i] based on (19, 36, 200)-net in base 27, using
- base change [i] based on digital (10, 27, 200)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 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, 18, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81 (see above)
- digital (1, 9, 100)-net over F81, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (10, 27, 200)-net over F81, using
(20, 20+17, 666)-Net over F27 — Digital
Digital (20, 37, 666)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2737, 666, F27, 17) (dual of [666, 629, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2737, 743, F27, 17) (dual of [743, 706, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(2733, 729, F27, 17) (dual of [729, 696, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2723, 729, F27, 12) (dual of [729, 706, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(16) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(2737, 743, F27, 17) (dual of [743, 706, 18]-code), using
(20, 20+17, 399806)-Net in Base 27 — Upper bound on s
There is no (20, 37, 399807)-net in base 27, because
- 1 times m-reduction [i] would yield (20, 36, 399807)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 3381 446329 094560 856948 707715 267639 813043 255665 892145 > 2736 [i]