Best Known (153−25, 153, s)-Nets in Base 3
(153−25, 153, 1643)-Net over F3 — Constructive and digital
Digital (128, 153, 1643)-net over F3, using
- net defined by OOA [i] based on linear OOA(3153, 1643, F3, 25, 25) (dual of [(1643, 25), 40922, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3153, 19717, F3, 25) (dual of [19717, 19564, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3153, 19718, F3, 25) (dual of [19718, 19565, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3118, 19683, F3, 20) (dual of [19683, 19565, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(38, 35, F3, 4) (dual of [35, 27, 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(24) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3153, 19718, F3, 25) (dual of [19718, 19565, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3153, 19717, F3, 25) (dual of [19717, 19564, 26]-code), using
(153−25, 153, 8502)-Net over F3 — Digital
Digital (128, 153, 8502)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3153, 8502, F3, 2, 25) (dual of [(8502, 2), 16851, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3153, 9859, F3, 2, 25) (dual of [(9859, 2), 19565, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3153, 19718, F3, 25) (dual of [19718, 19565, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3118, 19683, F3, 20) (dual of [19683, 19565, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(38, 35, F3, 4) (dual of [35, 27, 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(24) ⊂ Ce(19) [i] based on
- OOA 2-folding [i] based on linear OA(3153, 19718, F3, 25) (dual of [19718, 19565, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(3153, 9859, F3, 2, 25) (dual of [(9859, 2), 19565, 26]-NRT-code), using
(153−25, 153, 2923247)-Net in Base 3 — Upper bound on s
There is no (128, 153, 2923248)-net in base 3, because
- 1 times m-reduction [i] would yield (128, 152, 2923248)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3 329901 756706 578441 684766 112470 464643 715233 088893 658458 130417 078166 310017 > 3152 [i]