Best Known (129, 129+24, s)-Nets in Base 3
(129, 129+24, 1643)-Net over F3 — Constructive and digital
Digital (129, 153, 1643)-net over F3, using
- t-expansion [i] based on 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
- net defined by OOA [i] based on linear OOA(3153, 1643, F3, 25, 25) (dual of [(1643, 25), 40922, 26]-NRT-code), using
(129, 129+24, 9862)-Net over F3 — Digital
Digital (129, 153, 9862)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3153, 9862, F3, 2, 24) (dual of [(9862, 2), 19571, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3153, 19724, F3, 24) (dual of [19724, 19571, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3153, 19725, F3, 24) (dual of [19725, 19572, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3109, 19684, F3, 19) (dual of [19684, 19575, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3153, 19725, F3, 24) (dual of [19725, 19572, 25]-code), using
- OOA 2-folding [i] based on linear OA(3153, 19724, F3, 24) (dual of [19724, 19571, 25]-code), using
(129, 129+24, 3203508)-Net in Base 3 — Upper bound on s
There is no (129, 153, 3203509)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 9 989714 432534 503422 194366 394992 063701 317062 130304 892294 576594 157100 292017 > 3153 [i]