Best Known (60, 79, s)-Nets in Base 3
(60, 79, 328)-Net over F3 — Constructive and digital
Digital (60, 79, 328)-net over F3, using
- 1 times m-reduction [i] based on digital (60, 80, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 20, 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, 20, 82)-net over F81, using
(60, 79, 540)-Net over F3 — Digital
Digital (60, 79, 540)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(379, 540, F3, 19) (dual of [540, 461, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(379, 753, F3, 19) (dual of [753, 674, 20]-code), using
- construction XX applied to C1 = C([725,13]), C2 = C([1,15]), C3 = C1 + C2 = C([1,13]), and C∩ = C1 ∩ C2 = C([725,15]) [i] based on
- linear OA(367, 728, F3, 17) (dual of [728, 661, 18]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,13}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(360, 728, F3, 15) (dual of [728, 668, 16]-code), using the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(373, 728, F3, 19) (dual of [728, 655, 20]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,15}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(354, 728, F3, 13) (dual of [728, 674, 14]-code), using the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(35, 18, F3, 3) (dual of [18, 13, 4]-code or 18-cap in PG(4,3)), using
- linear OA(31, 7, F3, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([725,13]), C2 = C([1,15]), C3 = C1 + C2 = C([1,13]), and C∩ = C1 ∩ C2 = C([725,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(379, 753, F3, 19) (dual of [753, 674, 20]-code), using
(60, 79, 28290)-Net in Base 3 — Upper bound on s
There is no (60, 79, 28291)-net in base 3, because
- 1 times m-reduction [i] would yield (60, 78, 28291)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 16 425388 170788 201950 313110 165392 380599 > 378 [i]