Best Known (56, 56+22, s)-Nets in Base 3
(56, 56+22, 192)-Net over F3 — Constructive and digital
Digital (56, 78, 192)-net over F3, using
- trace code for nets [i] based on digital (4, 26, 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
(56, 56+22, 269)-Net over F3 — Digital
Digital (56, 78, 269)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(378, 269, F3, 22) (dual of [269, 191, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(377, 268, F3, 22) (dual of [268, 191, 23]-code), using
- construction XX applied to C1 = C([239,15]), C2 = C([0,18]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([239,18]) [i] based on
- 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(361, 242, F3, 19) (dual of [242, 181, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(371, 242, F3, 22) (dual of [242, 171, 23]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−3,−2,…,18}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(351, 242, F3, 16) (dual of [242, 191, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code) (see above)
- construction XX applied to C1 = C([239,15]), C2 = C([0,18]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([239,18]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(377, 268, F3, 22) (dual of [268, 191, 23]-code), using
(56, 56+22, 5921)-Net in Base 3 — Upper bound on s
There is no (56, 78, 5922)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 16 435509 729954 653701 772374 419735 041025 > 378 [i]