Best Known (66−32, 66, s)-Nets in Base 27
(66−32, 66, 178)-Net over F27 — Constructive and digital
Digital (34, 66, 178)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (8, 24, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- digital (10, 42, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- digital (8, 24, 84)-net over F27, using
(66−32, 66, 370)-Net in Base 27 — Constructive
(34, 66, 370)-net in base 27, using
- 6 times m-reduction [i] based on (34, 72, 370)-net in base 27, using
- base change [i] based on digital (16, 54, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 54, 370)-net over F81, using
(66−32, 66, 571)-Net over F27 — Digital
Digital (34, 66, 571)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2766, 571, F27, 32) (dual of [571, 505, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(2766, 746, F27, 32) (dual of [746, 680, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(24) [i] based on
- linear OA(2760, 729, F27, 32) (dual of [729, 669, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2749, 729, F27, 25) (dual of [729, 680, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(276, 17, F27, 6) (dual of [17, 11, 7]-code or 17-arc in PG(5,27)), using
- discarding factors / shortening the dual code based on linear OA(276, 27, F27, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,27)), using
- Reed–Solomon code RS(21,27) [i]
- discarding factors / shortening the dual code based on linear OA(276, 27, F27, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,27)), using
- construction X applied to Ce(31) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2766, 746, F27, 32) (dual of [746, 680, 33]-code), using
(66−32, 66, 209857)-Net in Base 27 — Upper bound on s
There is no (34, 66, 209858)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 29514 209274 711671 386778 700931 711649 950845 237863 148746 329487 742367 142240 810608 881245 985146 205345 > 2766 [i]