Best Known (48, 48+16, s)-Nets in Base 3
(48, 48+16, 328)-Net over F3 — Constructive and digital
Digital (48, 64, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 16, 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
(48, 48+16, 412)-Net over F3 — Digital
Digital (48, 64, 412)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(364, 412, F3, 16) (dual of [412, 348, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(364, 742, F3, 16) (dual of [742, 678, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(361, 729, F3, 16) (dual of [729, 668, 17]-code), using an extension Ce(15) of 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(349, 729, F3, 13) (dual of [729, 680, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(364, 742, F3, 16) (dual of [742, 678, 17]-code), using
(48, 48+16, 12341)-Net in Base 3 — Upper bound on s
There is no (48, 64, 12342)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3 434898 430564 302267 562946 122449 > 364 [i]