Best Known (68−36, 68, s)-Nets in Base 27
(68−36, 68, 164)-Net over F27 — Constructive and digital
Digital (32, 68, 164)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 25, 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 (7, 43, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27 (see above)
- digital (7, 25, 82)-net over F27, using
(68−36, 68, 224)-Net in Base 27 — Constructive
(32, 68, 224)-net in base 27, using
- 8 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
(68−36, 68, 353)-Net over F27 — Digital
Digital (32, 68, 353)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2768, 353, F27, 2, 36) (dual of [(353, 2), 638, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2768, 366, F27, 2, 36) (dual of [(366, 2), 664, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2768, 732, F27, 36) (dual of [732, 664, 37]-code), using
- construction XX applied to C1 = C([727,33]), C2 = C([0,34]), C3 = C1 + C2 = C([0,33]), and C∩ = C1 ∩ C2 = C([727,34]) [i] based on
- linear OA(2766, 728, F27, 35) (dual of [728, 662, 36]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,33}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2766, 728, F27, 35) (dual of [728, 662, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,34], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2768, 728, F27, 36) (dual of [728, 660, 37]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,34}, and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2764, 728, F27, 34) (dual of [728, 664, 35]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,33], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,33]), C2 = C([0,34]), C3 = C1 + C2 = C([0,33]), and C∩ = C1 ∩ C2 = C([727,34]) [i] based on
- OOA 2-folding [i] based on linear OA(2768, 732, F27, 36) (dual of [732, 664, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(2768, 366, F27, 2, 36) (dual of [(366, 2), 664, 37]-NRT-code), using
(68−36, 68, 74212)-Net in Base 27 — Upper bound on s
There is no (32, 68, 74213)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 21 517272 475321 181721 734235 062368 934522 899981 731405 585844 081942 805615 006499 303917 067413 427389 584929 > 2768 [i]