Best Known (162, 178, s)-Nets in Base 3
(162, 178, 1048689)-Net over F3 — Constructive and digital
Digital (162, 178, 1048689)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (19, 27, 114)-net over F3, using
- trace code for nets [i] based on digital (1, 9, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- trace code for nets [i] based on digital (1, 9, 38)-net over F27, using
- digital (135, 151, 1048575)-net over F3, using
- net defined by OOA [i] based on linear OOA(3151, 1048575, F3, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3151, 8388600, F3, 16) (dual of [8388600, 8388449, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3151, 8388600, F3, 16) (dual of [8388600, 8388449, 17]-code), using
- net defined by OOA [i] based on linear OOA(3151, 1048575, F3, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
- digital (19, 27, 114)-net over F3, using
(162, 178, 4194471)-Net over F3 — Digital
Digital (162, 178, 4194471)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3178, 4194471, F3, 2, 16) (dual of [(4194471, 2), 8388764, 17]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(327, 170, F3, 2, 8) (dual of [(170, 2), 313, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(327, 170, F3, 8) (dual of [170, 143, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(327, 253, F3, 8) (dual of [253, 226, 9]-code), using
- construction XX applied to C1 = C([115,121]), C2 = C([117,122]), C3 = C1 + C2 = C([117,121]), and C∩ = C1 ∩ C2 = C([115,122]) [i] based on
- linear OA(321, 242, F3, 7) (dual of [242, 221, 8]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {115,116,…,121}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(321, 242, F3, 6) (dual of [242, 221, 7]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {117,118,…,122}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(326, 242, F3, 8) (dual of [242, 216, 9]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {115,116,…,122}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(316, 242, F3, 5) (dual of [242, 226, 6]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {117,118,119,120,121}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- 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
- linear OA(30, 5, F3, 0) (dual of [5, 5, 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([115,121]), C2 = C([117,122]), C3 = C1 + C2 = C([117,121]), and C∩ = C1 ∩ C2 = C([115,122]) [i] based on
- discarding factors / shortening the dual code based on linear OA(327, 253, F3, 8) (dual of [253, 226, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(327, 170, F3, 8) (dual of [170, 143, 9]-code), using
- linear OOA(3151, 4194301, F3, 2, 16) (dual of [(4194301, 2), 8388451, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3151, 8388602, F3, 16) (dual of [8388602, 8388451, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- OOA 2-folding [i] based on linear OA(3151, 8388602, F3, 16) (dual of [8388602, 8388451, 17]-code), using
- linear OOA(327, 170, F3, 2, 8) (dual of [(170, 2), 313, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(162, 178, large)-Net in Base 3 — Upper bound on s
There is no (162, 178, large)-net in base 3, because
- 14 times m-reduction [i] would yield (162, 164, large)-net in base 3, but