Best Known (36, 72, s)-Nets in Base 27
(36, 72, 178)-Net over F27 — Constructive and digital
Digital (36, 72, 178)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (8, 26, 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, 46, 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, 26, 84)-net over F27, using
(36, 72, 370)-Net in Base 27 — Constructive
(36, 72, 370)-net in base 27, using
- 8 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, 72, 492)-Net over F27 — Digital
Digital (36, 72, 492)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2772, 492, F27, 36) (dual of [492, 420, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(2772, 743, F27, 36) (dual of [743, 671, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(30) [i] based on
- linear OA(2768, 729, F27, 36) (dual of [729, 661, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2758, 729, F27, 31) (dual of [729, 671, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(35) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(2772, 743, F27, 36) (dual of [743, 671, 37]-code), using
(36, 72, 154378)-Net in Base 27 — Upper bound on s
There is no (36, 72, 154379)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 11 434909 091434 376790 687290 471463 256221 652300 971107 331755 184925 852027 624983 430453 579376 785292 799146 567269 > 2772 [i]