Best Known (123, 141, s)-Nets in Base 3
(123, 141, 19687)-Net over F3 — Constructive and digital
Digital (123, 141, 19687)-net over F3, using
- t-expansion [i] based on digital (122, 141, 19687)-net over F3, using
- net defined by OOA [i] based on linear OOA(3141, 19687, F3, 19, 19) (dual of [(19687, 19), 373912, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3141, 177184, F3, 19) (dual of [177184, 177043, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3141, 177188, F3, 19) (dual of [177188, 177047, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3133, 177147, F3, 19) (dual of [177147, 177014, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3100, 177147, F3, 14) (dual of [177147, 177047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3141, 177188, F3, 19) (dual of [177188, 177047, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3141, 177184, F3, 19) (dual of [177184, 177043, 20]-code), using
- net defined by OOA [i] based on linear OOA(3141, 19687, F3, 19, 19) (dual of [(19687, 19), 373912, 20]-NRT-code), using
(123, 141, 84721)-Net over F3 — Digital
Digital (123, 141, 84721)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3141, 84721, F3, 2, 18) (dual of [(84721, 2), 169301, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3141, 88594, F3, 2, 18) (dual of [(88594, 2), 177047, 19]-NRT-code), using
- strength reduction [i] based on linear OOA(3141, 88594, F3, 2, 19) (dual of [(88594, 2), 177047, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3141, 177188, F3, 19) (dual of [177188, 177047, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3133, 177147, F3, 19) (dual of [177147, 177014, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3100, 177147, F3, 14) (dual of [177147, 177047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(3141, 177188, F3, 19) (dual of [177188, 177047, 20]-code), using
- strength reduction [i] based on linear OOA(3141, 88594, F3, 2, 19) (dual of [(88594, 2), 177047, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3141, 88594, F3, 2, 18) (dual of [(88594, 2), 177047, 19]-NRT-code), using
(123, 141, large)-Net in Base 3 — Upper bound on s
There is no (123, 141, large)-net in base 3, because
- 16 times m-reduction [i] would yield (123, 125, large)-net in base 3, but