Best Known (151−13, 151, s)-Nets in Base 3
(151−13, 151, 1398834)-Net over F3 — Constructive and digital
Digital (138, 151, 1398834)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (24, 30, 734)-net over F3, using
- net defined by OOA [i] based on linear OOA(330, 734, F3, 6, 6) (dual of [(734, 6), 4374, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(330, 734, F3, 5, 6) (dual of [(734, 5), 3640, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(330, 2202, F3, 6) (dual of [2202, 2172, 7]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(329, 2187, F3, 7) (dual of [2187, 2158, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(315, 2187, F3, 4) (dual of [2187, 2172, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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(6) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(330, 2202, F3, 6) (dual of [2202, 2172, 7]-code), using
- appending kth column [i] based on linear OOA(330, 734, F3, 5, 6) (dual of [(734, 5), 3640, 7]-NRT-code), using
- net defined by OOA [i] based on linear OOA(330, 734, F3, 6, 6) (dual of [(734, 6), 4374, 7]-NRT-code), using
- digital (108, 121, 1398100)-net over F3, using
- net defined by OOA [i] based on linear OOA(3121, 1398100, F3, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3121, 8388601, F3, 13) (dual of [8388601, 8388480, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3121, 8388601, F3, 13) (dual of [8388601, 8388480, 14]-code), using
- net defined by OOA [i] based on linear OOA(3121, 1398100, F3, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
- digital (24, 30, 734)-net over F3, using
(151−13, 151, 7873767)-Net over F3 — Digital
Digital (138, 151, 7873767)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3151, 7873767, F3, 13) (dual of [7873767, 7873616, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 13) (dual of [large, large−151, 14]-code), using
- strength reduction [i] based on linear OA(3151, large, F3, 15) (dual of [large, large−151, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- strength reduction [i] based on linear OA(3151, large, F3, 15) (dual of [large, large−151, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 13) (dual of [large, large−151, 14]-code), using
(151−13, 151, large)-Net in Base 3 — Upper bound on s
There is no (138, 151, large)-net in base 3, because
- 11 times m-reduction [i] would yield (138, 140, large)-net in base 3, but