Best Known (53, 53+13, s)-Nets in Base 128
(53, 53+13, 2097153)-Net over F128 — Constructive and digital
Digital (53, 66, 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 (36, 49, 1398100)-net over F128, using
- net defined by OOA [i] based on linear OOA(12849, 1398100, F128, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12849, 8388601, F128, 13) (dual of [8388601, 8388552, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12849, large, F128, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, 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(12849, large, F128, 13) (dual of [large, large−49, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12849, 8388601, F128, 13) (dual of [8388601, 8388552, 14]-code), using
- net defined by OOA [i] based on linear OOA(12849, 1398100, F128, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- digital (11, 17, 699053)-net over F128, using
(53, 53+13, 2796329)-Net in Base 128 — Constructive
(53, 66, 2796329)-net in base 128, using
- net defined by OOA [i] based on OOA(12866, 2796329, S128, 15, 13), using
- OOA 2-folding and stacking with additional row [i] based on OOA(12866, 5592659, S128, 3, 13), using
- discarding parts of the base [i] based on linear OOA(25657, 5592659, F256, 3, 13) (dual of [(5592659, 3), 16777920, 14]-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(25637, 2796201, F256, 3, 13) (dual of [(2796201, 3), 8388566, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- OOA 3-folding [i] based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-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(25657, 5592659, F256, 3, 13) (dual of [(5592659, 3), 16777920, 14]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on OOA(12866, 5592659, S128, 3, 13), using
(53, 53+13, large)-Net over F128 — Digital
Digital (53, 66, large)-net over F128, using
- t-expansion [i] based on digital (51, 66, large)-net over F128, using
- 4 times m-reduction [i] based on digital (51, 70, large)-net over F128, using
(53, 53+13, large)-Net in Base 128 — Upper bound on s
There is no (53, 66, large)-net in base 128, because
- 11 times m-reduction [i] would yield (53, 55, large)-net in base 128, but