Best Known (45, 45+22, s)-Nets in Base 128
(45, 45+22, 190651)-Net over F128 — Constructive and digital
Digital (45, 67, 190651)-net over F128, using
- 1281 times duplication [i] based on digital (44, 66, 190651)-net over F128, using
- net defined by OOA [i] based on linear OOA(12866, 190651, F128, 22, 22) (dual of [(190651, 22), 4194256, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(12866, 2097161, F128, 22) (dual of [2097161, 2097095, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(12866, 2097163, F128, 22) (dual of [2097163, 2097097, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- 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(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(12866, 2097163, F128, 22) (dual of [2097163, 2097097, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(12866, 2097161, F128, 22) (dual of [2097161, 2097095, 23]-code), using
- net defined by OOA [i] based on linear OOA(12866, 190651, F128, 22, 22) (dual of [(190651, 22), 4194256, 23]-NRT-code), using
(45, 45+22, 1009914)-Net over F128 — Digital
Digital (45, 67, 1009914)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12867, 1009914, F128, 2, 22) (dual of [(1009914, 2), 2019761, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12867, 1048583, F128, 2, 22) (dual of [(1048583, 2), 2097099, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12867, 2097166, F128, 22) (dual of [2097166, 2097099, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(12867, 2097167, F128, 22) (dual of [2097167, 2097100, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- 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(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(12867, 2097167, F128, 22) (dual of [2097167, 2097100, 23]-code), using
- OOA 2-folding [i] based on linear OA(12867, 2097166, F128, 22) (dual of [2097166, 2097099, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(12867, 1048583, F128, 2, 22) (dual of [(1048583, 2), 2097099, 23]-NRT-code), using
(45, 45+22, large)-Net in Base 128 — Upper bound on s
There is no (45, 67, large)-net in base 128, because
- 20 times m-reduction [i] would yield (45, 47, large)-net in base 128, but