Best Known (87, 87+19, s)-Nets in Base 3
(87, 87+19, 733)-Net over F3 — Constructive and digital
Digital (87, 106, 733)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (78, 97, 729)-net over F3, using
- net defined by OOA [i] based on linear OOA(397, 729, F3, 19, 19) (dual of [(729, 19), 13754, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using
- net defined by OOA [i] based on linear OOA(397, 729, F3, 19, 19) (dual of [(729, 19), 13754, 20]-NRT-code), using
- digital (0, 9, 4)-net over F3, using
(87, 87+19, 3297)-Net over F3 — Digital
Digital (87, 106, 3297)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3106, 3297, F3, 2, 19) (dual of [(3297, 2), 6488, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3106, 6594, F3, 19) (dual of [6594, 6488, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3106, 6595, F3, 19) (dual of [6595, 6489, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(365, 6561, F3, 13) (dual of [6561, 6496, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(38, 33, F3, 4) (dual of [33, 25, 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
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(18) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(3106, 6595, F3, 19) (dual of [6595, 6489, 20]-code), using
- OOA 2-folding [i] based on linear OA(3106, 6594, F3, 19) (dual of [6594, 6488, 20]-code), using
(87, 87+19, 764066)-Net in Base 3 — Upper bound on s
There is no (87, 106, 764067)-net in base 3, because
- 1 times m-reduction [i] would yield (87, 105, 764067)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 125 237213 643704 896681 991064 597388 376621 242116 220663 > 3105 [i]