Best Known (30, 30+29, s)-Nets in Base 128
(30, 30+29, 1170)-Net over F128 — Constructive and digital
Digital (30, 59, 1170)-net over F128, using
- 1282 times duplication [i] based on digital (28, 57, 1170)-net over F128, using
- net defined by OOA [i] based on linear OOA(12857, 1170, F128, 29, 29) (dual of [(1170, 29), 33873, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(12857, 16381, F128, 29) (dual of [16381, 16324, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(12857, 16384, F128, 29) (dual of [16384, 16327, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(12857, 16384, F128, 29) (dual of [16384, 16327, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(12857, 16381, F128, 29) (dual of [16381, 16324, 30]-code), using
- net defined by OOA [i] based on linear OOA(12857, 1170, F128, 29, 29) (dual of [(1170, 29), 33873, 30]-NRT-code), using
(30, 30+29, 4195)-Net over F128 — Digital
Digital (30, 59, 4195)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12859, 4195, F128, 3, 29) (dual of [(4195, 3), 12526, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12859, 5464, F128, 3, 29) (dual of [(5464, 3), 16333, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12859, 16392, F128, 29) (dual of [16392, 16333, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- linear OA(12857, 16384, F128, 29) (dual of [16384, 16327, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-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(28) ⊂ Ce(25) [i] based on
- OOA 3-folding [i] based on linear OA(12859, 16392, F128, 29) (dual of [16392, 16333, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(12859, 5464, F128, 3, 29) (dual of [(5464, 3), 16333, 30]-NRT-code), using
(30, 30+29, large)-Net in Base 128 — Upper bound on s
There is no (30, 59, large)-net in base 128, because
- 27 times m-reduction [i] would yield (30, 32, large)-net in base 128, but