Best Known (67−28, 67, s)-Nets in Base 128
(67−28, 67, 1173)-Net over F128 — Constructive and digital
Digital (39, 67, 1173)-net over F128, using
- net defined by OOA [i] based on linear OOA(12867, 1173, F128, 28, 28) (dual of [(1173, 28), 32777, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(12867, 16422, F128, 28) (dual of [16422, 16355, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(14) [i] based on
- linear OA(12855, 16384, F128, 28) (dual of [16384, 16329, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(12829, 16384, F128, 15) (dual of [16384, 16355, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(12812, 38, F128, 12) (dual of [38, 26, 13]-code or 38-arc in PG(11,128)), using
- discarding factors / shortening the dual code based on linear OA(12812, 128, F128, 12) (dual of [128, 116, 13]-code or 128-arc in PG(11,128)), using
- Reed–Solomon code RS(116,128) [i]
- discarding factors / shortening the dual code based on linear OA(12812, 128, F128, 12) (dual of [128, 116, 13]-code or 128-arc in PG(11,128)), using
- construction X applied to Ce(27) ⊂ Ce(14) [i] based on
- OA 14-folding and stacking [i] based on linear OA(12867, 16422, F128, 28) (dual of [16422, 16355, 29]-code), using
(67−28, 67, 4681)-Net in Base 128 — Constructive
(39, 67, 4681)-net in base 128, using
- 1281 times duplication [i] based on (38, 66, 4681)-net in base 128, using
- t-expansion [i] based on (37, 66, 4681)-net in base 128, using
- net defined by OOA [i] based on OOA(12866, 4681, S128, 29, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(12866, 65535, S128, 29), using
- discarding factors based on OA(12866, 65538, S128, 29), using
- discarding parts of the base [i] based on linear OA(25657, 65538, F256, 29) (dual of [65538, 65481, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- 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(25655, 65536, F256, 28) (dual of [65536, 65481, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(28) ⊂ Ce(27) [i] based on
- discarding parts of the base [i] based on linear OA(25657, 65538, F256, 29) (dual of [65538, 65481, 30]-code), using
- discarding factors based on OA(12866, 65538, S128, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(12866, 65535, S128, 29), using
- net defined by OOA [i] based on OOA(12866, 4681, S128, 29, 29), using
- t-expansion [i] based on (37, 66, 4681)-net in base 128, using
(67−28, 67, 16422)-Net over F128 — Digital
Digital (39, 67, 16422)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12867, 16422, F128, 28) (dual of [16422, 16355, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(14) [i] based on
- linear OA(12855, 16384, F128, 28) (dual of [16384, 16329, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(12829, 16384, F128, 15) (dual of [16384, 16355, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(12812, 38, F128, 12) (dual of [38, 26, 13]-code or 38-arc in PG(11,128)), using
- discarding factors / shortening the dual code based on linear OA(12812, 128, F128, 12) (dual of [128, 116, 13]-code or 128-arc in PG(11,128)), using
- Reed–Solomon code RS(116,128) [i]
- discarding factors / shortening the dual code based on linear OA(12812, 128, F128, 12) (dual of [128, 116, 13]-code or 128-arc in PG(11,128)), using
- construction X applied to Ce(27) ⊂ Ce(14) [i] based on
(67−28, 67, large)-Net in Base 128 — Upper bound on s
There is no (39, 67, large)-net in base 128, because
- 26 times m-reduction [i] would yield (39, 41, large)-net in base 128, but