Best Known (48, 65, s)-Nets in Base 3
(48, 65, 192)-Net over F3 — Constructive and digital
Digital (48, 65, 192)-net over F3, using
- 1 times m-reduction [i] based on digital (48, 66, 192)-net over F3, using
- trace code for nets [i] based on digital (4, 22, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- trace code for nets [i] based on digital (4, 22, 64)-net over F27, using
(48, 65, 336)-Net over F3 — Digital
Digital (48, 65, 336)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(365, 336, F3, 17) (dual of [336, 271, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(365, 374, F3, 17) (dual of [374, 309, 18]-code), using
- construction XX applied to C1 = C([167,182]), C2 = C([169,183]), C3 = C1 + C2 = C([169,182]), and C∩ = C1 ∩ C2 = C([167,183]) [i] based on
- linear OA(358, 364, F3, 16) (dual of [364, 306, 17]-code), using the BCH-code C(I) with length 364 | 36−1, defining interval I = {167,168,…,182}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(361, 364, F3, 15) (dual of [364, 303, 16]-code), using the BCH-code C(I) with length 364 | 36−1, defining interval I = {169,170,…,183}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(364, 364, F3, 17) (dual of [364, 300, 18]-code), using the BCH-code C(I) with length 364 | 36−1, defining interval I = {167,168,…,183}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(355, 364, F3, 14) (dual of [364, 309, 15]-code), using the BCH-code C(I) with length 364 | 36−1, defining interval I = {169,170,…,182}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(31, 4, F3, 1) (dual of [4, 3, 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
- 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
- construction XX applied to C1 = C([167,182]), C2 = C([169,183]), C3 = C1 + C2 = C([169,182]), and C∩ = C1 ∩ C2 = C([167,183]) [i] based on
- discarding factors / shortening the dual code based on linear OA(365, 374, F3, 17) (dual of [374, 309, 18]-code), using
(48, 65, 12341)-Net in Base 3 — Upper bound on s
There is no (48, 65, 12342)-net in base 3, because
- 1 times m-reduction [i] would yield (48, 64, 12342)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3 434898 430564 302267 562946 122449 > 364 [i]