Best Known (33−16, 33, s)-Nets in Base 27
(33−16, 33, 132)-Net over F27 — Constructive and digital
Digital (17, 33, 132)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (5, 21, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- digital (4, 12, 64)-net over F27, using
(33−16, 33, 172)-Net in Base 27 — Constructive
(17, 33, 172)-net in base 27, using
- 7 times m-reduction [i] based on (17, 40, 172)-net in base 27, using
- base change [i] based on digital (7, 30, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- base change [i] based on digital (7, 30, 172)-net over F81, using
(33−16, 33, 429)-Net over F27 — Digital
Digital (17, 33, 429)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2733, 429, F27, 16) (dual of [429, 396, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2733, 737, F27, 16) (dual of [737, 704, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(2731, 729, F27, 16) (dual of [729, 698, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(2725, 729, F27, 13) (dual of [729, 704, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(2733, 737, F27, 16) (dual of [737, 704, 17]-code), using
(33−16, 33, 116165)-Net in Base 27 — Upper bound on s
There is no (17, 33, 116166)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 171795 752775 660617 239636 990474 452986 124901 839825 > 2733 [i]