Best Known (40, 40+21, s)-Nets in Base 128
(40, 40+21, 209715)-Net over F128 — Constructive and digital
Digital (40, 61, 209715)-net over F128, using
- net defined by OOA [i] based on linear OOA(12861, 209715, F128, 21, 21) (dual of [(209715, 21), 4403954, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12861, 2097151, F128, 21) (dual of [2097151, 2097090, 22]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12861, 2097151, F128, 21) (dual of [2097151, 2097090, 22]-code), using
(40, 40+21, 699051)-Net over F128 — Digital
Digital (40, 61, 699051)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12861, 699051, F128, 3, 21) (dual of [(699051, 3), 2097092, 22]-NRT-code), using
- OOA 3-folding [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]
- OOA 3-folding [i] based on linear OA(12861, 2097153, F128, 21) (dual of [2097153, 2097092, 22]-code), using
(40, 40+21, large)-Net in Base 128 — Upper bound on s
There is no (40, 61, large)-net in base 128, because
- 19 times m-reduction [i] would yield (40, 42, large)-net in base 128, but