Best Known (137, 137+13, s)-Nets in Base 3
(137, 137+13, 1398831)-Net over F3 — Constructive and digital
Digital (137, 150, 1398831)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (23, 29, 731)-net over F3, using
- net defined by OOA [i] based on linear OOA(329, 731, F3, 6, 6) (dual of [(731, 6), 4357, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(329, 731, F3, 5, 6) (dual of [(731, 5), 3626, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(329, 2193, F3, 6) (dual of [2193, 2164, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(329, 2195, F3, 6) (dual of [2195, 2166, 7]-code), using
- 1 times truncation [i] based on linear OA(330, 2196, F3, 7) (dual of [2196, 2166, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [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(322, 2187, F3, 5) (dual of [2187, 2165, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(38, 9, F3, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,3)), using
- dual of repetition code with length 9 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 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 X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- 1 times truncation [i] based on linear OA(330, 2196, F3, 7) (dual of [2196, 2166, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(329, 2195, F3, 6) (dual of [2195, 2166, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(329, 2193, F3, 6) (dual of [2193, 2164, 7]-code), using
- appending kth column [i] based on linear OOA(329, 731, F3, 5, 6) (dual of [(731, 5), 3626, 7]-NRT-code), using
- net defined by OOA [i] based on linear OOA(329, 731, F3, 6, 6) (dual of [(731, 6), 4357, 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 (23, 29, 731)-net over F3, using
(137, 137+13, 7125377)-Net over F3 — Digital
Digital (137, 150, 7125377)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3150, 7125377, F3, 13) (dual of [7125377, 7125227, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, large, F3, 13) (dual of [large, large−150, 14]-code), using
- strength reduction [i] based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- strength reduction [i] based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, large, F3, 13) (dual of [large, large−150, 14]-code), using
(137, 137+13, large)-Net in Base 3 — Upper bound on s
There is no (137, 150, large)-net in base 3, because
- 11 times m-reduction [i] would yield (137, 139, large)-net in base 3, but