Best Known (34, 65, s)-Nets in Base 27
(34, 65, 182)-Net over F27 — Constructive and digital
Digital (34, 65, 182)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (9, 24, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- digital (10, 41, 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 (9, 24, 88)-net over F27, using
(34, 65, 370)-Net in Base 27 — Constructive
(34, 65, 370)-net in base 27, using
- 7 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
(34, 65, 634)-Net over F27 — Digital
Digital (34, 65, 634)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2765, 634, F27, 31) (dual of [634, 569, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2765, 749, F27, 31) (dual of [749, 684, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- 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(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(277, 20, F27, 7) (dual of [20, 13, 8]-code or 20-arc in PG(6,27)), using
- discarding factors / shortening the dual code based on linear OA(277, 27, F27, 7) (dual of [27, 20, 8]-code or 27-arc in PG(6,27)), using
- Reed–Solomon code RS(20,27) [i]
- discarding factors / shortening the dual code based on linear OA(277, 27, F27, 7) (dual of [27, 20, 8]-code or 27-arc in PG(6,27)), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2765, 749, F27, 31) (dual of [749, 684, 32]-code), using
(34, 65, 316180)-Net in Base 27 — Upper bound on s
There is no (34, 65, 316181)-net in base 27, because
- 1 times m-reduction [i] would yield (34, 64, 316181)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 40 484916 387594 739660 922420 100399 921953 454220 690579 532440 029055 703128 826905 468097 968017 077955 > 2764 [i]