Best Known (190, 190+34, s)-Nets in Base 3
(190, 190+34, 3474)-Net over F3 — Constructive and digital
Digital (190, 224, 3474)-net over F3, using
- 32 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
(190, 190+34, 19675)-Net over F3 — Digital
Digital (190, 224, 19675)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3224, 19675, F3, 3, 34) (dual of [(19675, 3), 58801, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3224, 19687, F3, 3, 34) (dual of [(19687, 3), 58837, 35]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3223, 19687, F3, 3, 34) (dual of [(19687, 3), 58838, 35]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3223, 59061, F3, 34) (dual of [59061, 58838, 35]-code), using
- 1 times code embedding in larger space [i] 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
- 1 times code embedding in larger space [i] based on linear OA(3222, 59060, F3, 34) (dual of [59060, 58838, 35]-code), using
- OOA 3-folding [i] based on linear OA(3223, 59061, F3, 34) (dual of [59061, 58838, 35]-code), using
- 31 times duplication [i] based on linear OOA(3223, 19687, F3, 3, 34) (dual of [(19687, 3), 58838, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3224, 19687, F3, 3, 34) (dual of [(19687, 3), 58837, 35]-NRT-code), using
(190, 190+34, 6945257)-Net in Base 3 — Upper bound on s
There is no (190, 224, 6945258)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 75017 306843 928922 860244 723833 620770 823082 324608 581566 837378 742006 986109 064347 569097 152034 348906 392096 024117 > 3224 [i]