Best Known (57−28, 57, s)-Nets in Base 256
(57−28, 57, 4681)-Net over F256 — Constructive and digital
Digital (29, 57, 4681)-net over F256, using
- t-expansion [i] based on digital (28, 57, 4681)-net over F256, using
- net defined by OOA [i] based on linear OOA(25657, 4681, F256, 29, 29) (dual of [(4681, 29), 135692, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(25657, 65535, F256, 29) (dual of [65535, 65478, 30]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(25657, 65536, F256, 29) (dual of [65536, 65479, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(25657, 65535, F256, 29) (dual of [65535, 65478, 30]-code), using
- net defined by OOA [i] based on linear OOA(25657, 4681, F256, 29, 29) (dual of [(4681, 29), 135692, 30]-NRT-code), using
(57−28, 57, 13108)-Net over F256 — Digital
Digital (29, 57, 13108)-net over F256, using
- 2561 times duplication [i] based on digital (28, 56, 13108)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25656, 13108, F256, 5, 28) (dual of [(13108, 5), 65484, 29]-NRT-code), using
- OOA 5-folding [i] based on linear OA(25656, 65540, F256, 28) (dual of [65540, 65484, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(25656, 65541, F256, 28) (dual of [65541, 65485, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [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(25651, 65536, F256, 26) (dual of [65536, 65485, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 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(25656, 65541, F256, 28) (dual of [65541, 65485, 29]-code), using
- OOA 5-folding [i] based on linear OA(25656, 65540, F256, 28) (dual of [65540, 65484, 29]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25656, 13108, F256, 5, 28) (dual of [(13108, 5), 65484, 29]-NRT-code), using
(57−28, 57, large)-Net in Base 256 — Upper bound on s
There is no (29, 57, large)-net in base 256, because
- 26 times m-reduction [i] would yield (29, 31, large)-net in base 256, but