Best Known (56−28, 56, s)-Nets in Base 128
(56−28, 56, 1170)-Net over F128 — Constructive and digital
Digital (28, 56, 1170)-net over F128, using
- 1 times m-reduction [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
(56−28, 56, 4097)-Net over F128 — Digital
Digital (28, 56, 4097)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12856, 4097, F128, 4, 28) (dual of [(4097, 4), 16332, 29]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12856, 16388, F128, 28) (dual of [16388, 16332, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(12856, 16389, F128, 28) (dual of [16389, 16333, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [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(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(1281, 5, F128, 1) (dual of [5, 4, 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(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(12856, 16389, F128, 28) (dual of [16389, 16333, 29]-code), using
- OOA 4-folding [i] based on linear OA(12856, 16388, F128, 28) (dual of [16388, 16332, 29]-code), using
(56−28, 56, large)-Net in Base 128 — Upper bound on s
There is no (28, 56, large)-net in base 128, because
- 26 times m-reduction [i] would yield (28, 30, large)-net in base 128, but