Best Known (257−29, 257, s)-Nets in Base 2
(257−29, 257, 18726)-Net over F2 — Constructive and digital
Digital (228, 257, 18726)-net over F2, using
- 21 times duplication [i] based on digital (227, 256, 18726)-net over F2, using
- net defined by OOA [i] based on linear OOA(2256, 18726, F2, 29, 29) (dual of [(18726, 29), 542798, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2256, 262165, F2, 29) (dual of [262165, 261909, 30]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2254, 262163, F2, 29) (dual of [262163, 261909, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2253, 262144, F2, 29) (dual of [262144, 261891, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2235, 262144, F2, 27) (dual of [262144, 261909, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(21, 19, F2, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2254, 262163, F2, 29) (dual of [262163, 261909, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2256, 262165, F2, 29) (dual of [262165, 261909, 30]-code), using
- net defined by OOA [i] based on linear OOA(2256, 18726, F2, 29, 29) (dual of [(18726, 29), 542798, 30]-NRT-code), using
(257−29, 257, 33249)-Net over F2 — Digital
Digital (228, 257, 33249)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2257, 33249, F2, 7, 29) (dual of [(33249, 7), 232486, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2257, 37452, F2, 7, 29) (dual of [(37452, 7), 261907, 30]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2255, 37452, F2, 7, 29) (dual of [(37452, 7), 261909, 30]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2255, 262164, F2, 29) (dual of [262164, 261909, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2254, 262163, F2, 29) (dual of [262163, 261909, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2253, 262144, F2, 29) (dual of [262144, 261891, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2235, 262144, F2, 27) (dual of [262144, 261909, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(21, 19, F2, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2254, 262163, F2, 29) (dual of [262163, 261909, 30]-code), using
- OOA 7-folding [i] based on linear OA(2255, 262164, F2, 29) (dual of [262164, 261909, 30]-code), using
- 22 times duplication [i] based on linear OOA(2255, 37452, F2, 7, 29) (dual of [(37452, 7), 261909, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2257, 37452, F2, 7, 29) (dual of [(37452, 7), 261907, 30]-NRT-code), using
(257−29, 257, 1931975)-Net in Base 2 — Upper bound on s
There is no (228, 257, 1931976)-net in base 2, because
- 1 times m-reduction [i] would yield (228, 256, 1931976)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 115792 476380 107112 409933 410196 265889 808373 690683 723072 680118 571557 171948 532104 > 2256 [i]