Best Known (129−19, 129, s)-Nets in Base 3
(129−19, 129, 6565)-Net over F3 — Constructive and digital
Digital (110, 129, 6565)-net over F3, using
- net defined by OOA [i] based on linear OOA(3129, 6565, F3, 19, 19) (dual of [(6565, 19), 124606, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3129, 59086, F3, 19) (dual of [59086, 58957, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3129, 59087, F3, 19) (dual of [59087, 58958, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(391, 59049, F3, 14) (dual of [59049, 58958, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(38, 38, F3, 4) (dual of [38, 30, 5]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3129, 59087, F3, 19) (dual of [59087, 58958, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3129, 59086, F3, 19) (dual of [59086, 58957, 20]-code), using
(129−19, 129, 20814)-Net over F3 — Digital
Digital (110, 129, 20814)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3129, 20814, F3, 2, 19) (dual of [(20814, 2), 41499, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3129, 29543, F3, 2, 19) (dual of [(29543, 2), 58957, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3129, 59086, F3, 19) (dual of [59086, 58957, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3129, 59087, F3, 19) (dual of [59087, 58958, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(391, 59049, F3, 14) (dual of [59049, 58958, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(38, 38, F3, 4) (dual of [38, 30, 5]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3129, 59087, F3, 19) (dual of [59087, 58958, 20]-code), using
- OOA 2-folding [i] based on linear OA(3129, 59086, F3, 19) (dual of [59086, 58957, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(3129, 29543, F3, 2, 19) (dual of [(29543, 2), 58957, 20]-NRT-code), using
(129−19, 129, large)-Net in Base 3 — Upper bound on s
There is no (110, 129, large)-net in base 3, because
- 17 times m-reduction [i] would yield (110, 112, large)-net in base 3, but