Best Known (258−35, 258, s)-Nets in Base 2
(258−35, 258, 1928)-Net over F2 — Constructive and digital
Digital (223, 258, 1928)-net over F2, using
- 21 times duplication [i] based on digital (222, 257, 1928)-net over F2, using
- net defined by OOA [i] based on linear OOA(2257, 1928, F2, 35, 35) (dual of [(1928, 35), 67223, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2257, 32777, F2, 35) (dual of [32777, 32520, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2257, 32784, F2, 35) (dual of [32784, 32527, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- linear OA(2256, 32768, F2, 35) (dual of [32768, 32512, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2241, 32768, F2, 33) (dual of [32768, 32527, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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(34) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(2257, 32784, F2, 35) (dual of [32784, 32527, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2257, 32777, F2, 35) (dual of [32777, 32520, 36]-code), using
- net defined by OOA [i] based on linear OOA(2257, 1928, F2, 35, 35) (dual of [(1928, 35), 67223, 36]-NRT-code), using
(258−35, 258, 5464)-Net over F2 — Digital
Digital (223, 258, 5464)-net over F2, using
- 21 times duplication [i] based on digital (222, 257, 5464)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2257, 5464, F2, 6, 35) (dual of [(5464, 6), 32527, 36]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2257, 32784, F2, 35) (dual of [32784, 32527, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- linear OA(2256, 32768, F2, 35) (dual of [32768, 32512, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2241, 32768, F2, 33) (dual of [32768, 32527, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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(34) ⊂ Ce(32) [i] based on
- OOA 6-folding [i] based on linear OA(2257, 32784, F2, 35) (dual of [32784, 32527, 36]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2257, 5464, F2, 6, 35) (dual of [(5464, 6), 32527, 36]-NRT-code), using
(258−35, 258, 255133)-Net in Base 2 — Upper bound on s
There is no (223, 258, 255134)-net in base 2, because
- 1 times m-reduction [i] would yield (223, 257, 255134)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 231588 295814 441760 240700 847978 673922 031604 654167 444382 615513 513770 898435 115725 > 2257 [i]