Best Known (48, 60, s)-Nets in Base 3
(48, 60, 464)-Net over F3 — Constructive and digital
Digital (48, 60, 464)-net over F3, using
- t-expansion [i] based on digital (47, 60, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 15, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 15, 116)-net over F81, using
(48, 60, 1470)-Net over F3 — Digital
Digital (48, 60, 1470)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(360, 1470, F3, 12) (dual of [1470, 1410, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(360, 2201, F3, 12) (dual of [2201, 2141, 13]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(357, 2188, F3, 13) (dual of [2188, 2131, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(343, 2188, F3, 9) (dual of [2188, 2145, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(360, 2201, F3, 12) (dual of [2201, 2141, 13]-code), using
(48, 60, 88384)-Net in Base 3 — Upper bound on s
There is no (48, 60, 88385)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 42391 717936 676859 410873 864025 > 360 [i]