Best Known (70−21, 70, s)-Nets in Base 128
(70−21, 70, 209719)-Net over F128 — Constructive and digital
Digital (49, 70, 209719)-net over F128, using
- net defined by OOA [i] based on linear OOA(12870, 209719, F128, 21, 21) (dual of [(209719, 21), 4404029, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12870, 2097191, F128, 21) (dual of [2097191, 2097121, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12870, 2097192, F128, 21) (dual of [2097192, 2097122, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,5]) [i] based on
- linear OA(12861, 2097153, F128, 21) (dual of [2097153, 2097092, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(1289, 39, F128, 9) (dual of [39, 30, 10]-code or 39-arc in PG(8,128)), using
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- Reed–Solomon code RS(119,128) [i]
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- construction X applied to C([0,10]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12870, 2097192, F128, 21) (dual of [2097192, 2097122, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12870, 2097191, F128, 21) (dual of [2097191, 2097121, 22]-code), using
(70−21, 70, 838860)-Net in Base 128 — Constructive
(49, 70, 838860)-net in base 128, using
- net defined by OOA [i] based on OOA(12870, 838860, S128, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(12870, 8388601, S128, 21), using
- discarding factors based on OA(12870, large, S128, 21), using
- discarding parts of the base [i] based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding parts of the base [i] based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- discarding factors based on OA(12870, large, S128, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(12870, 8388601, S128, 21), using
(70−21, 70, 2097192)-Net over F128 — Digital
Digital (49, 70, 2097192)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12870, 2097192, F128, 21) (dual of [2097192, 2097122, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,5]) [i] based on
- linear OA(12861, 2097153, F128, 21) (dual of [2097153, 2097092, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(1289, 39, F128, 9) (dual of [39, 30, 10]-code or 39-arc in PG(8,128)), using
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- Reed–Solomon code RS(119,128) [i]
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- construction X applied to C([0,10]) ⊂ C([0,5]) [i] based on
(70−21, 70, large)-Net in Base 128 — Upper bound on s
There is no (49, 70, large)-net in base 128, because
- 19 times m-reduction [i] would yield (49, 51, large)-net in base 128, but