Best Known (42−21, 42, s)-Nets in Base 128
(42−21, 42, 1638)-Net over F128 — Constructive and digital
Digital (21, 42, 1638)-net over F128, using
- 1281 times duplication [i] based on digital (20, 41, 1638)-net over F128, using
- net defined by OOA [i] based on linear OOA(12841, 1638, F128, 21, 21) (dual of [(1638, 21), 34357, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12841, 16381, F128, 21) (dual of [16381, 16340, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12841, 16381, F128, 21) (dual of [16381, 16340, 22]-code), using
- net defined by OOA [i] based on linear OOA(12841, 1638, F128, 21, 21) (dual of [(1638, 21), 34357, 22]-NRT-code), using
(42−21, 42, 4097)-Net over F128 — Digital
Digital (21, 42, 4097)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12842, 4097, F128, 4, 21) (dual of [(4097, 4), 16346, 22]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12842, 16388, F128, 21) (dual of [16388, 16346, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12842, 16390, F128, 21) (dual of [16390, 16348, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(12841, 16385, F128, 21) (dual of [16385, 16344, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(12837, 16385, F128, 19) (dual of [16385, 16348, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 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(12842, 16390, F128, 21) (dual of [16390, 16348, 22]-code), using
- OOA 4-folding [i] based on linear OA(12842, 16388, F128, 21) (dual of [16388, 16346, 22]-code), using
(42−21, 42, large)-Net in Base 128 — Upper bound on s
There is no (21, 42, large)-net in base 128, because
- 19 times m-reduction [i] would yield (21, 23, large)-net in base 128, but