Best Known (36, 68, s)-Nets in Base 27
(36, 68, 188)-Net over F27 — Constructive and digital
Digital (36, 68, 188)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (10, 26, 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 (10, 42, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27 (see above)
- digital (10, 26, 94)-net over F27, using
(36, 68, 370)-Net in Base 27 — Constructive
(36, 68, 370)-net in base 27, using
- 12 times m-reduction [i] based on (36, 80, 370)-net in base 27, using
- base change [i] based on digital (16, 60, 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, 60, 370)-net over F81, using
(36, 68, 715)-Net over F27 — Digital
Digital (36, 68, 715)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2768, 715, F27, 32) (dual of [715, 647, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(2768, 752, F27, 32) (dual of [752, 684, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(22) [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(2745, 729, F27, 23) (dual of [729, 684, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(278, 23, F27, 8) (dual of [23, 15, 9]-code or 23-arc in PG(7,27)), using
- discarding factors / shortening the dual code based on linear OA(278, 27, F27, 8) (dual of [27, 19, 9]-code or 27-arc in PG(7,27)), using
- Reed–Solomon code RS(19,27) [i]
- discarding factors / shortening the dual code based on linear OA(278, 27, F27, 8) (dual of [27, 19, 9]-code or 27-arc in PG(7,27)), using
- construction X applied to Ce(31) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2768, 752, F27, 32) (dual of [752, 684, 33]-code), using
(36, 68, 316847)-Net in Base 27 — Upper bound on s
There is no (36, 68, 316848)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 21 515303 109835 974009 991231 089109 654779 101381 080406 854115 110437 169238 440623 894740 624298 222168 747009 > 2768 [i]