Best Known (41, 41+21, s)-Nets in Base 128
(41, 41+21, 209715)-Net over F128 — Constructive and digital
Digital (41, 62, 209715)-net over F128, using
- 1281 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(12861, 209715, F128, 21, 21) (dual of [(209715, 21), 4403954, 22]-NRT-code), using
(41, 41+21, 699053)-Net over F128 — Digital
Digital (41, 62, 699053)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12862, 699053, F128, 3, 21) (dual of [(699053, 3), 2097097, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12862, 2097159, F128, 21) (dual of [2097159, 2097097, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12862, 2097160, F128, 21) (dual of [2097160, 2097098, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [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(12855, 2097153, F128, 19) (dual of [2097153, 2097098, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12862, 2097160, F128, 21) (dual of [2097160, 2097098, 22]-code), using
- OOA 3-folding [i] based on linear OA(12862, 2097159, F128, 21) (dual of [2097159, 2097097, 22]-code), using
(41, 41+21, large)-Net in Base 128 — Upper bound on s
There is no (41, 62, large)-net in base 128, because
- 19 times m-reduction [i] would yield (41, 43, large)-net in base 128, but