Best Known (228, 228+22, s)-Nets in Base 3
(228, 228+22, 762744)-Net over F3 — Constructive and digital
Digital (228, 250, 762744)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (28, 39, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 13, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- trace code for nets [i] based on digital (2, 13, 48)-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 (28, 39, 144)-net over F3, using
(228, 228+22, 4194508)-Net over F3 — Digital
Digital (228, 250, 4194508)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3250, 4194508, F3, 2, 22) (dual of [(4194508, 2), 8388766, 23]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(339, 207, F3, 2, 11) (dual of [(207, 2), 375, 12]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(339, 207, F3, 11) (dual of [207, 168, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(339, 256, F3, 11) (dual of [256, 217, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(336, 243, F3, 11) (dual of [243, 207, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(326, 243, F3, 8) (dual of [243, 217, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(339, 256, F3, 11) (dual of [256, 217, 12]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(339, 207, F3, 11) (dual of [207, 168, 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(339, 207, F3, 2, 11) (dual of [(207, 2), 375, 12]-NRT-code), using
- (u, u+v)-construction [i] based on
(228, 228+22, large)-Net in Base 3 — Upper bound on s
There is no (228, 250, large)-net in base 3, because
- 20 times m-reduction [i] would yield (228, 230, large)-net in base 3, but