Best Known (70−37, 70, s)-Nets in Base 27
(70−37, 70, 166)-Net over F27 — Constructive and digital
Digital (33, 70, 166)-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 (8, 45, 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, 25, 82)-net over F27, using
(70−37, 70, 224)-Net in Base 27 — Constructive
(33, 70, 224)-net in base 27, using
- 10 times m-reduction [i] based on (33, 80, 224)-net in base 27, using
- base change [i] based on digital (13, 60, 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, 60, 224)-net over F81, using
(70−37, 70, 363)-Net over F27 — Digital
Digital (33, 70, 363)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2770, 363, F27, 2, 37) (dual of [(363, 2), 656, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2770, 366, F27, 2, 37) (dual of [(366, 2), 662, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2770, 732, F27, 37) (dual of [732, 662, 38]-code), using
- construction XX applied to C1 = C([727,34]), C2 = C([0,35]), C3 = C1 + C2 = C([0,34]), and C∩ = C1 ∩ C2 = C([727,35]) [i] based on
- 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(2768, 728, F27, 36) (dual of [728, 660, 37]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,35], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2770, 728, F27, 37) (dual of [728, 658, 38]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,35}, and designed minimum distance d ≥ |I|+1 = 38 [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(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([727,34]), C2 = C([0,35]), C3 = C1 + C2 = C([0,34]), and C∩ = C1 ∩ C2 = C([727,35]) [i] based on
- OOA 2-folding [i] based on linear OA(2770, 732, F27, 37) (dual of [732, 662, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(2770, 366, F27, 2, 37) (dual of [(366, 2), 662, 38]-NRT-code), using
(70−37, 70, 89126)-Net in Base 27 — Upper bound on s
There is no (33, 70, 89127)-net in base 27, because
- 1 times m-reduction [i] would yield (33, 69, 89127)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 580 968558 720985 225922 163306 792071 797451 412791 701908 433106 752899 213536 632572 894108 990991 878831 328765 > 2769 [i]