Best Known (48, 48+24, s)-Nets in Base 128
(48, 48+24, 174763)-Net over F128 — Constructive and digital
Digital (48, 72, 174763)-net over F128, using
- 1281 times duplication [i] based on digital (47, 71, 174763)-net over F128, using
- net defined by OOA [i] based on linear OOA(12871, 174763, F128, 24, 24) (dual of [(174763, 24), 4194241, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(12871, 2097156, F128, 24) (dual of [2097156, 2097085, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(12871, 2097159, F128, 24) (dual of [2097159, 2097088, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12864, 2097152, F128, 22) (dual of [2097152, 2097088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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 Ce(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(12871, 2097159, F128, 24) (dual of [2097159, 2097088, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(12871, 2097156, F128, 24) (dual of [2097156, 2097085, 25]-code), using
- net defined by OOA [i] based on linear OOA(12871, 174763, F128, 24, 24) (dual of [(174763, 24), 4194241, 25]-NRT-code), using
(48, 48+24, 722298)-Net over F128 — Digital
Digital (48, 72, 722298)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12872, 722298, F128, 2, 24) (dual of [(722298, 2), 1444524, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12872, 1048581, F128, 2, 24) (dual of [(1048581, 2), 2097090, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12872, 2097162, F128, 24) (dual of [2097162, 2097090, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(12872, 2097163, F128, 24) (dual of [2097163, 2097091, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- 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(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(23) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(12872, 2097163, F128, 24) (dual of [2097163, 2097091, 25]-code), using
- OOA 2-folding [i] based on linear OA(12872, 2097162, F128, 24) (dual of [2097162, 2097090, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(12872, 1048581, F128, 2, 24) (dual of [(1048581, 2), 2097090, 25]-NRT-code), using
(48, 48+24, large)-Net in Base 128 — Upper bound on s
There is no (48, 72, large)-net in base 128, because
- 22 times m-reduction [i] would yield (48, 50, large)-net in base 128, but