Best Known (27, 27+28, s)-Nets in Base 256
(27, 27+28, 4681)-Net over F256 — Constructive and digital
Digital (27, 55, 4681)-net over F256, using
- net defined by OOA [i] based on linear OOA(25655, 4681, F256, 28, 28) (dual of [(4681, 28), 131013, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(25655, 65534, F256, 28) (dual of [65534, 65479, 29]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(25655, 65536, F256, 28) (dual of [65536, 65481, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(25655, 65534, F256, 28) (dual of [65534, 65479, 29]-code), using
(27, 27+28, 10923)-Net over F256 — Digital
Digital (27, 55, 10923)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25655, 10923, F256, 6, 28) (dual of [(10923, 6), 65483, 29]-NRT-code), using
- OOA 6-folding [i] based on linear OA(25655, 65538, F256, 28) (dual of [65538, 65483, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(26) [i] based on
- 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(25653, 65536, F256, 27) (dual of [65536, 65483, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(27) ⊂ Ce(26) [i] based on
- OOA 6-folding [i] based on linear OA(25655, 65538, F256, 28) (dual of [65538, 65483, 29]-code), using
(27, 27+28, large)-Net in Base 256 — Upper bound on s
There is no (27, 55, large)-net in base 256, because
- 26 times m-reduction [i] would yield (27, 29, large)-net in base 256, but