Best Known (193−19, 193, s)-Nets in Base 3
(193−19, 193, 932076)-Net over F3 — Constructive and digital
Digital (174, 193, 932076)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- digital (162, 181, 932066)-net over F3, using
- net defined by OOA [i] based on linear OOA(3181, 932066, F3, 19, 19) (dual of [(932066, 19), 17709073, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3181, 8388595, F3, 19) (dual of [8388595, 8388414, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3181, 8388595, F3, 19) (dual of [8388595, 8388414, 20]-code), using
- net defined by OOA [i] based on linear OOA(3181, 932066, F3, 19, 19) (dual of [(932066, 19), 17709073, 20]-NRT-code), using
- digital (3, 12, 10)-net over F3, using
(193−19, 193, 2796211)-Net over F3 — Digital
Digital (174, 193, 2796211)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3193, 2796211, F3, 3, 19) (dual of [(2796211, 3), 8388440, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(312, 10, F3, 3, 9) (dual of [(10, 3), 18, 10]-NRT-code), using
- extended algebraic-geometric NRT-code AGe(3;F,20P) [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- linear OOA(3181, 2796201, F3, 3, 19) (dual of [(2796201, 3), 8388422, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- linear OOA(312, 10, F3, 3, 9) (dual of [(10, 3), 18, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(193−19, 193, large)-Net in Base 3 — Upper bound on s
There is no (174, 193, large)-net in base 3, because
- 17 times m-reduction [i] would yield (174, 176, large)-net in base 3, but