Best Known (63−18, 63, s)-Nets in Base 3
(63−18, 63, 156)-Net over F3 — Constructive and digital
Digital (45, 63, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 21, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
(63−18, 63, 227)-Net over F3 — Digital
Digital (45, 63, 227)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(363, 227, F3, 18) (dual of [227, 164, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(363, 248, F3, 18) (dual of [248, 185, 19]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(361, 244, F3, 19) (dual of [244, 183, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 244 | 310−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(351, 244, F3, 15) (dual of [244, 193, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 244 | 310−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(32, 4, F3, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
- extended Reed–Solomon code RSe(2,3) [i]
- Hamming code H(2,3) [i]
- Simplex code S(2,3) [i]
- the Tetracode [i]
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(363, 248, F3, 18) (dual of [248, 185, 19]-code), using
(63−18, 63, 4526)-Net in Base 3 — Upper bound on s
There is no (45, 63, 4527)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 145849 847373 615196 643327 369551 > 363 [i]