Best Known (73−19, 73, s)-Nets in Base 128
(73−19, 73, 932066)-Net over F128 — Constructive and digital
Digital (54, 73, 932066)-net over F128, using
- net defined by OOA [i] based on linear OOA(12873, 932066, F128, 19, 19) (dual of [(932066, 19), 17709181, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12873, 8388595, F128, 19) (dual of [8388595, 8388522, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12873, large, F128, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, 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(12873, large, F128, 19) (dual of [large, large−73, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12873, 8388595, F128, 19) (dual of [8388595, 8388522, 20]-code), using
(73−19, 73, 932216)-Net in Base 128 — Constructive
(54, 73, 932216)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- (44, 63, 932066)-net in base 128, using
- net defined by OOA [i] based on OOA(12863, 932066, S128, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12863, 8388595, S128, 19), using
- discarding factors based on OA(12863, large, S128, 19), using
- discarding parts of the base [i] based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding parts of the base [i] based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- discarding factors based on OA(12863, large, S128, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12863, 8388595, S128, 19), using
- net defined by OOA [i] based on OOA(12863, 932066, S128, 19, 19), using
- digital (1, 10, 150)-net over F128, using
(73−19, 73, large)-Net over F128 — Digital
Digital (54, 73, large)-net over F128, using
- 1 times m-reduction [i] based on digital (54, 74, large)-net over F128, using
(73−19, 73, large)-Net in Base 128 — Upper bound on s
There is no (54, 73, large)-net in base 128, because
- 17 times m-reduction [i] would yield (54, 56, large)-net in base 128, but