Best Known (20, 20+6, s)-Nets in Base 3
(20, 20+6, 328)-Net over F3 — Constructive and digital
Digital (20, 26, 328)-net over F3, using
- 32 times duplication [i] based on digital (18, 24, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 6, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 6, 82)-net over F81, using
(20, 20+6, 745)-Net over F3 — Digital
Digital (20, 26, 745)-net over F3, using
- net defined by OOA [i] based on linear OOA(326, 745, F3, 6, 6) (dual of [(745, 6), 4444, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(326, 745, F3, 5, 6) (dual of [(745, 5), 3699, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(326, 745, F3, 6) (dual of [745, 719, 7]-code), using
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(325, 740, F3, 6) (dual of [740, 715, 7]-code), using
- construction XX applied to C1 = C([727,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([727,4]) [i] based on
- linear OA(319, 728, F3, 5) (dual of [728, 709, 6]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,1,2,3}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(319, 728, F3, 5) (dual of [728, 709, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(325, 728, F3, 6) (dual of [728, 703, 7]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(313, 728, F3, 4) (dual of [728, 715, 5]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code) (see above)
- construction XX applied to C1 = C([727,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([727,4]) [i] based on
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(325, 740, F3, 6) (dual of [740, 715, 7]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(326, 745, F3, 6) (dual of [745, 719, 7]-code), using
- appending kth column [i] based on linear OOA(326, 745, F3, 5, 6) (dual of [(745, 5), 3699, 7]-NRT-code), using
(20, 20+6, 12397)-Net in Base 3 — Upper bound on s
There is no (20, 26, 12398)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2 542472 737601 > 326 [i]