Best Known (64−21, 64, s)-Nets in Base 128
(64−21, 64, 209716)-Net over F128 — Constructive and digital
Digital (43, 64, 209716)-net over F128, using
- 1281 times duplication [i] based on digital (42, 63, 209716)-net over F128, using
- net defined by OOA [i] based on linear OOA(12863, 209716, F128, 21, 21) (dual of [(209716, 21), 4403973, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12863, 2097161, F128, 21) (dual of [2097161, 2097098, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12863, 2097163, F128, 21) (dual of [2097163, 2097100, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(12863, 2097163, F128, 21) (dual of [2097163, 2097100, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12863, 2097161, F128, 21) (dual of [2097161, 2097098, 22]-code), using
- net defined by OOA [i] based on linear OOA(12863, 209716, F128, 21, 21) (dual of [(209716, 21), 4403973, 22]-NRT-code), using
(64−21, 64, 1048584)-Net over F128 — Digital
Digital (43, 64, 1048584)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12864, 1048584, F128, 2, 21) (dual of [(1048584, 2), 2097104, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12864, 2097168, F128, 21) (dual of [2097168, 2097104, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [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(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- OOA 2-folding [i] based on linear OA(12864, 2097168, F128, 21) (dual of [2097168, 2097104, 22]-code), using
(64−21, 64, large)-Net in Base 128 — Upper bound on s
There is no (43, 64, large)-net in base 128, because
- 19 times m-reduction [i] would yield (43, 45, large)-net in base 128, but