Best Known (226, 226+22, s)-Nets in Base 3
(226, 226+22, 762714)-Net over F3 — Constructive and digital
Digital (226, 248, 762714)-net over F3, using
- 31 times duplication [i] based on digital (225, 247, 762714)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (25, 36, 114)-net over F3, using
- trace code for nets [i] based on digital (1, 12, 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, 12, 38)-net over F27, using
- digital (189, 211, 762600)-net over F3, using
- net defined by OOA [i] based on linear OOA(3211, 762600, F3, 22, 22) (dual of [(762600, 22), 16776989, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3211, 8388600, F3, 22) (dual of [8388600, 8388389, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3211, large, F3, 22) (dual of [large, large−211, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(3211, large, F3, 22) (dual of [large, large−211, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3211, 8388600, F3, 22) (dual of [8388600, 8388389, 23]-code), using
- net defined by OOA [i] based on linear OOA(3211, 762600, F3, 22, 22) (dual of [(762600, 22), 16776989, 23]-NRT-code), using
- digital (25, 36, 114)-net over F3, using
- (u, u+v)-construction [i] based on
(226, 226+22, 4194462)-Net over F3 — Digital
Digital (226, 248, 4194462)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3248, 4194462, F3, 2, 22) (dual of [(4194462, 2), 8388676, 23]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(337, 161, F3, 2, 11) (dual of [(161, 2), 285, 12]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(337, 161, F3, 11) (dual of [161, 124, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(337, 253, F3, 11) (dual of [253, 216, 12]-code), using
- construction XX applied to C1 = C([112,121]), C2 = C([114,122]), C3 = C1 + C2 = C([114,121]), and C∩ = C1 ∩ C2 = C([112,122]) [i] based on
- linear OA(331, 242, F3, 10) (dual of [242, 211, 11]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {112,113,…,121}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(331, 242, F3, 9) (dual of [242, 211, 10]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {114,115,…,122}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(336, 242, F3, 11) (dual of [242, 206, 12]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {112,113,…,122}, and designed minimum distance d ≥ |I|+1 = 12 [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 = {114,115,…,121}, and designed minimum distance d ≥ |I|+1 = 9 [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([112,121]), C2 = C([114,122]), C3 = C1 + C2 = C([114,121]), and C∩ = C1 ∩ C2 = C([112,122]) [i] based on
- discarding factors / shortening the dual code based on linear OA(337, 253, F3, 11) (dual of [253, 216, 12]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(337, 161, F3, 11) (dual of [161, 124, 12]-code), using
- linear OOA(3211, 4194301, F3, 2, 22) (dual of [(4194301, 2), 8388391, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3211, 8388602, F3, 22) (dual of [8388602, 8388391, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3211, large, F3, 22) (dual of [large, large−211, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(3211, large, F3, 22) (dual of [large, large−211, 23]-code), using
- OOA 2-folding [i] based on linear OA(3211, 8388602, F3, 22) (dual of [8388602, 8388391, 23]-code), using
- linear OOA(337, 161, F3, 2, 11) (dual of [(161, 2), 285, 12]-NRT-code), using
- (u, u+v)-construction [i] based on
(226, 226+22, large)-Net in Base 3 — Upper bound on s
There is no (226, 248, large)-net in base 3, because
- 20 times m-reduction [i] would yield (226, 228, large)-net in base 3, but