Best Known (52−26, 52, s)-Nets in Base 128
(52−26, 52, 1260)-Net over F128 — Constructive and digital
Digital (26, 52, 1260)-net over F128, using
- 1 times m-reduction [i] based on digital (26, 53, 1260)-net over F128, using
- net defined by OOA [i] based on linear OOA(12853, 1260, F128, 27, 27) (dual of [(1260, 27), 33967, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12853, 16381, F128, 27) (dual of [16381, 16328, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(12853, 16384, F128, 27) (dual of [16384, 16331, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(12853, 16384, F128, 27) (dual of [16384, 16331, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12853, 16381, F128, 27) (dual of [16381, 16328, 28]-code), using
- net defined by OOA [i] based on linear OOA(12853, 1260, F128, 27, 27) (dual of [(1260, 27), 33967, 28]-NRT-code), using
(52−26, 52, 4097)-Net over F128 — Digital
Digital (26, 52, 4097)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12852, 4097, F128, 4, 26) (dual of [(4097, 4), 16336, 27]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12852, 16388, F128, 26) (dual of [16388, 16336, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(12852, 16389, F128, 26) (dual of [16389, 16337, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- 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(12847, 16384, F128, 24) (dual of [16384, 16337, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(25) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(12852, 16389, F128, 26) (dual of [16389, 16337, 27]-code), using
- OOA 4-folding [i] based on linear OA(12852, 16388, F128, 26) (dual of [16388, 16336, 27]-code), using
(52−26, 52, large)-Net in Base 128 — Upper bound on s
There is no (26, 52, large)-net in base 128, because
- 24 times m-reduction [i] would yield (26, 28, large)-net in base 128, but