Best Known (48, 48+19, s)-Nets in Base 3
(48, 48+19, 156)-Net over F3 — Constructive and digital
Digital (48, 67, 156)-net over F3, using
- 31 times duplication [i] based on digital (47, 66, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 22, 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
- trace code for nets [i] based on digital (3, 22, 52)-net over F27, using
(48, 48+19, 242)-Net over F3 — Digital
Digital (48, 67, 242)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(367, 242, F3, 19) (dual of [242, 175, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(367, 264, F3, 19) (dual of [264, 197, 20]-code), using
- construction XX applied to C1 = C([239,13]), C2 = C([1,15]), C3 = C1 + C2 = C([1,13]), and C∩ = C1 ∩ C2 = C([239,15]) [i] based on
- linear OA(356, 242, F3, 17) (dual of [242, 186, 18]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−3,−2,…,13}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(350, 242, F3, 15) (dual of [242, 192, 16]-code), using the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(361, 242, F3, 19) (dual of [242, 181, 20]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−3,−2,…,15}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(345, 242, F3, 13) (dual of [242, 197, 14]-code), using the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(35, 16, F3, 3) (dual of [16, 11, 4]-code or 16-cap in PG(4,3)), using
- linear OA(31, 6, F3, 1) (dual of [6, 5, 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([239,13]), C2 = C([1,15]), C3 = C1 + C2 = C([1,13]), and C∩ = C1 ∩ C2 = C([239,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(367, 264, F3, 19) (dual of [264, 197, 20]-code), using
(48, 48+19, 6532)-Net in Base 3 — Upper bound on s
There is no (48, 67, 6533)-net in base 3, because
- 1 times m-reduction [i] would yield (48, 66, 6533)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 30 945697 128481 623365 349739 770475 > 366 [i]