Best Known (172, 172+30, s)-Nets in Base 3
(172, 172+30, 3938)-Net over F3 — Constructive and digital
Digital (172, 202, 3938)-net over F3, using
- net defined by OOA [i] based on linear OOA(3202, 3938, F3, 30, 30) (dual of [(3938, 30), 117938, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(3202, 59070, F3, 30) (dual of [59070, 58868, 31]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- 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(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(31, 21, F3, 1) (dual of [21, 20, 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(30) ⊂ Ce(27) [i] based on
- OA 15-folding and stacking [i] based on linear OA(3202, 59070, F3, 30) (dual of [59070, 58868, 31]-code), using
(172, 172+30, 19690)-Net over F3 — Digital
Digital (172, 202, 19690)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3202, 19690, F3, 3, 30) (dual of [(19690, 3), 58868, 31]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3202, 59070, F3, 30) (dual of [59070, 58868, 31]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- 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(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(31, 21, F3, 1) (dual of [21, 20, 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(30) ⊂ Ce(27) [i] based on
- OOA 3-folding [i] based on linear OA(3202, 59070, F3, 30) (dual of [59070, 58868, 31]-code), using
(172, 172+30, large)-Net in Base 3 — Upper bound on s
There is no (172, 202, large)-net in base 3, because
- 28 times m-reduction [i] would yield (172, 174, large)-net in base 3, but