Best Known (32, 67, s)-Nets in Base 27
(32, 67, 166)-Net over F27 — Constructive and digital
Digital (32, 67, 166)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 24, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (8, 43, 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 (7, 24, 82)-net over F27, using
(32, 67, 224)-Net in Base 27 — Constructive
(32, 67, 224)-net in base 27, using
- 9 times m-reduction [i] based on (32, 76, 224)-net in base 27, using
- base change [i] based on digital (13, 57, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- base change [i] based on digital (13, 57, 224)-net over F81, using
(32, 67, 367)-Net over F27 — Digital
Digital (32, 67, 367)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2767, 367, F27, 2, 35) (dual of [(367, 2), 667, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2767, 734, F27, 35) (dual of [734, 667, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- linear OA(2766, 729, F27, 35) (dual of [729, 663, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2762, 729, F27, 33) (dual of [729, 667, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- OOA 2-folding [i] based on linear OA(2767, 734, F27, 35) (dual of [734, 667, 36]-code), using
(32, 67, 99539)-Net in Base 27 — Upper bound on s
There is no (32, 67, 99540)-net in base 27, because
- 1 times m-reduction [i] would yield (32, 66, 99540)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 29517 532152 801614 091677 559816 839988 420554 520951 492558 570162 462747 782794 878973 432978 204565 577353 > 2766 [i]