Best Known (50, 50+12, s)-Nets in Base 128
(50, 50+12, 2097153)-Net over F128 — Constructive and digital
Digital (50, 62, 2097153)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (11, 17, 699053)-net over F128, using
- net defined by OOA [i] based on linear OOA(12817, 699053, F128, 6, 6) (dual of [(699053, 6), 4194301, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(12817, 2097159, F128, 6) (dual of [2097159, 2097142, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(12817, 2097159, F128, 6) (dual of [2097159, 2097142, 7]-code), using
- net defined by OOA [i] based on linear OOA(12817, 699053, F128, 6, 6) (dual of [(699053, 6), 4194301, 7]-NRT-code), using
- digital (33, 45, 1398100)-net over F128, using
- net defined by OOA [i] based on linear OOA(12845, 1398100, F128, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(12845, 8388600, F128, 12) (dual of [8388600, 8388555, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(12845, large, F128, 12) (dual of [large, large−45, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9256395 | 1284−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(12845, large, F128, 12) (dual of [large, large−45, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(12845, 8388600, F128, 12) (dual of [8388600, 8388555, 13]-code), using
- net defined by OOA [i] based on linear OOA(12845, 1398100, F128, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- digital (11, 17, 699053)-net over F128, using
(50, 50+12, 2796329)-Net in Base 128 — Constructive
(50, 62, 2796329)-net in base 128, using
- net defined by OOA [i] based on OOA(12862, 2796329, S128, 15, 12), using
- OOA 2-folding and stacking with additional row [i] based on OOA(12862, 5592659, S128, 3, 12), using
- discarding parts of the base [i] based on linear OOA(25654, 5592659, F256, 3, 12) (dual of [(5592659, 3), 16777923, 13]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(2564, 257, F256, 3, 4) (dual of [(257, 3), 767, 5]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;767,256) [i]
- linear OOA(25616, 2796201, F256, 3, 6) (dual of [(2796201, 3), 8388587, 7]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- linear OOA(25634, 2796201, F256, 3, 12) (dual of [(2796201, 3), 8388569, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 3-folding [i] based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- linear OOA(2564, 257, F256, 3, 4) (dual of [(257, 3), 767, 5]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- discarding parts of the base [i] based on linear OOA(25654, 5592659, F256, 3, 12) (dual of [(5592659, 3), 16777923, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on OOA(12862, 5592659, S128, 3, 12), using
(50, 50+12, large)-Net over F128 — Digital
Digital (50, 62, large)-net over F128, using
- t-expansion [i] based on digital (48, 62, large)-net over F128, using
- 4 times m-reduction [i] based on digital (48, 66, large)-net over F128, using
(50, 50+12, large)-Net in Base 128 — Upper bound on s
There is no (50, 62, large)-net in base 128, because
- 10 times m-reduction [i] would yield (50, 52, large)-net in base 128, but