Best Known (25, 50, s)-Nets in Base 128
(25, 50, 1365)-Net over F128 — Constructive and digital
Digital (25, 50, 1365)-net over F128, using
- 1281 times duplication [i] based on digital (24, 49, 1365)-net over F128, using
- net defined by OOA [i] based on linear OOA(12849, 1365, F128, 25, 25) (dual of [(1365, 25), 34076, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12849, 16381, F128, 25) (dual of [16381, 16332, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12849, 16384, F128, 25) (dual of [16384, 16335, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(12849, 16384, F128, 25) (dual of [16384, 16335, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12849, 16381, F128, 25) (dual of [16381, 16332, 26]-code), using
- net defined by OOA [i] based on linear OOA(12849, 1365, F128, 25, 25) (dual of [(1365, 25), 34076, 26]-NRT-code), using
(25, 50, 4097)-Net over F128 — Digital
Digital (25, 50, 4097)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12850, 4097, F128, 4, 25) (dual of [(4097, 4), 16338, 26]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12850, 16388, F128, 25) (dual of [16388, 16338, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12850, 16390, F128, 25) (dual of [16390, 16340, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(12849, 16385, F128, 25) (dual of [16385, 16336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(12845, 16385, F128, 23) (dual of [16385, 16340, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12850, 16390, F128, 25) (dual of [16390, 16340, 26]-code), using
- OOA 4-folding [i] based on linear OA(12850, 16388, F128, 25) (dual of [16388, 16338, 26]-code), using
(25, 50, large)-Net in Base 128 — Upper bound on s
There is no (25, 50, large)-net in base 128, because
- 23 times m-reduction [i] would yield (25, 27, large)-net in base 128, but