Best Known (249−32, 249, s)-Nets in Base 3
(249−32, 249, 11079)-Net over F3 — Constructive and digital
Digital (217, 249, 11079)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 17, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (200, 232, 11072)-net over F3, using
- net defined by OOA [i] based on linear OOA(3232, 11072, F3, 32, 32) (dual of [(11072, 32), 354072, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(3232, 177152, F3, 32) (dual of [177152, 176920, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3232, 177158, F3, 32) (dual of [177158, 176926, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(30) [i] based on
- linear OA(3232, 177147, F3, 32) (dual of [177147, 176915, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3221, 177147, F3, 31) (dual of [177147, 176926, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 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 X applied to Ce(31) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3232, 177158, F3, 32) (dual of [177158, 176926, 33]-code), using
- OA 16-folding and stacking [i] based on linear OA(3232, 177152, F3, 32) (dual of [177152, 176920, 33]-code), using
- net defined by OOA [i] based on linear OOA(3232, 11072, F3, 32, 32) (dual of [(11072, 32), 354072, 33]-NRT-code), using
- digital (1, 17, 7)-net over F3, using
(249−32, 249, 67557)-Net over F3 — Digital
Digital (217, 249, 67557)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3249, 67557, F3, 2, 32) (dual of [(67557, 2), 134865, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3249, 88609, F3, 2, 32) (dual of [(88609, 2), 176969, 33]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3248, 88609, F3, 2, 32) (dual of [(88609, 2), 176970, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3248, 177218, F3, 32) (dual of [177218, 176970, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(24) [i] based on
- linear OA(3232, 177147, F3, 32) (dual of [177147, 176915, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3177, 177147, F3, 25) (dual of [177147, 176970, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(316, 71, F3, 6) (dual of [71, 55, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 80, F3, 6) (dual of [80, 64, 7]-code), using
- the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(316, 80, F3, 6) (dual of [80, 64, 7]-code), using
- construction X applied to Ce(31) ⊂ Ce(24) [i] based on
- OOA 2-folding [i] based on linear OA(3248, 177218, F3, 32) (dual of [177218, 176970, 33]-code), using
- 31 times duplication [i] based on linear OOA(3248, 88609, F3, 2, 32) (dual of [(88609, 2), 176970, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3249, 88609, F3, 2, 32) (dual of [(88609, 2), 176969, 33]-NRT-code), using
(249−32, 249, large)-Net in Base 3 — Upper bound on s
There is no (217, 249, large)-net in base 3, because
- 30 times m-reduction [i] would yield (217, 219, large)-net in base 3, but