Best Known (226−34, 226, s)-Nets in Base 3
(226−34, 226, 3474)-Net over F3 — Constructive and digital
Digital (192, 226, 3474)-net over F3, using
- 34 times duplication [i] based on digital (188, 222, 3474)-net over F3, using
- net defined by OOA [i] based on linear OOA(3222, 3474, F3, 34, 34) (dual of [(3474, 34), 117894, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(3222, 59058, F3, 34) (dual of [59058, 58836, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3222, 59060, F3, 34) (dual of [59060, 58838, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3211, 59049, F3, 32) (dual of [59049, 58838, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3222, 59060, F3, 34) (dual of [59060, 58838, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(3222, 59058, F3, 34) (dual of [59058, 58836, 35]-code), using
- net defined by OOA [i] based on linear OOA(3222, 3474, F3, 34, 34) (dual of [(3474, 34), 117894, 35]-NRT-code), using
(226−34, 226, 19691)-Net over F3 — Digital
Digital (192, 226, 19691)-net over F3, using
- 31 times duplication [i] based on digital (191, 225, 19691)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3225, 19691, F3, 3, 34) (dual of [(19691, 3), 58848, 35]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3225, 59073, F3, 34) (dual of [59073, 58848, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- OOA 3-folding [i] based on linear OA(3225, 59073, F3, 34) (dual of [59073, 58848, 35]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3225, 19691, F3, 3, 34) (dual of [(19691, 3), 58848, 35]-NRT-code), using
(226−34, 226, 7903517)-Net in Base 3 — Upper bound on s
There is no (192, 226, 7903518)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 675156 446450 649810 344892 456530 935036 864097 609772 093288 586151 901006 764027 924714 771696 428923 027412 679303 044125 > 3226 [i]