Best Known (68−30, 68, s)-Nets in Base 128
(68−30, 68, 1094)-Net over F128 — Constructive and digital
Digital (38, 68, 1094)-net over F128, using
- 1281 times duplication [i] based on digital (37, 67, 1094)-net over F128, using
- net defined by OOA [i] based on linear OOA(12867, 1094, F128, 30, 30) (dual of [(1094, 30), 32753, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(12867, 16410, F128, 30) (dual of [16410, 16343, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(20) [i] based on
- linear OA(12859, 16384, F128, 30) (dual of [16384, 16325, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- 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]
- linear OA(1288, 26, F128, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,128)), using
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- Reed–Solomon code RS(120,128) [i]
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- construction X applied to Ce(29) ⊂ Ce(20) [i] based on
- OA 15-folding and stacking [i] based on linear OA(12867, 16410, F128, 30) (dual of [16410, 16343, 31]-code), using
- net defined by OOA [i] based on linear OOA(12867, 1094, F128, 30, 30) (dual of [(1094, 30), 32753, 31]-NRT-code), using
(68−30, 68, 4369)-Net in Base 128 — Constructive
(38, 68, 4369)-net in base 128, using
- net defined by OOA [i] based on OOA(12868, 4369, S128, 30, 30), using
- OA 15-folding and stacking [i] based on OA(12868, 65535, S128, 30), using
- discarding factors based on OA(12868, 65538, S128, 30), using
- discarding parts of the base [i] based on linear OA(25659, 65538, F256, 30) (dual of [65538, 65479, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(25659, 65536, F256, 30) (dual of [65536, 65477, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(25657, 65536, F256, 29) (dual of [65536, 65479, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- discarding parts of the base [i] based on linear OA(25659, 65538, F256, 30) (dual of [65538, 65479, 31]-code), using
- discarding factors based on OA(12868, 65538, S128, 30), using
- OA 15-folding and stacking [i] based on OA(12868, 65535, S128, 30), using
(68−30, 68, 9792)-Net over F128 — Digital
Digital (38, 68, 9792)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12868, 9792, F128, 30) (dual of [9792, 9724, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(12868, 16413, F128, 30) (dual of [16413, 16345, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(19) [i] based on
- linear OA(12859, 16384, F128, 30) (dual of [16384, 16325, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1289, 29, F128, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,128)), using
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- Reed–Solomon code RS(119,128) [i]
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- construction X applied to Ce(29) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(12868, 16413, F128, 30) (dual of [16413, 16345, 31]-code), using
(68−30, 68, large)-Net in Base 128 — Upper bound on s
There is no (38, 68, large)-net in base 128, because
- 28 times m-reduction [i] would yield (38, 40, large)-net in base 128, but