Best Known (257−25, 257, s)-Nets in Base 2
(257−25, 257, 174764)-Net over F2 — Constructive and digital
Digital (232, 257, 174764)-net over F2, using
- 23 times duplication [i] based on digital (229, 254, 174764)-net over F2, using
- net defined by OOA [i] based on linear OOA(2254, 174764, F2, 25, 25) (dual of [(174764, 25), 4368846, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2254, 2097169, F2, 25) (dual of [2097169, 2096915, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2254, 2097174, F2, 25) (dual of [2097174, 2096920, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2253, 2097152, F2, 25) (dual of [2097152, 2096899, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2232, 2097152, F2, 23) (dual of [2097152, 2096920, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2254, 2097174, F2, 25) (dual of [2097174, 2096920, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2254, 2097169, F2, 25) (dual of [2097169, 2096915, 26]-code), using
- net defined by OOA [i] based on linear OOA(2254, 174764, F2, 25, 25) (dual of [(174764, 25), 4368846, 26]-NRT-code), using
(257−25, 257, 262147)-Net over F2 — Digital
Digital (232, 257, 262147)-net over F2, using
- 21 times duplication [i] based on digital (231, 256, 262147)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2256, 262147, F2, 8, 25) (dual of [(262147, 8), 2096920, 26]-NRT-code), using
- OOA 8-folding [i] based on linear OA(2256, 2097176, F2, 25) (dual of [2097176, 2096920, 26]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2254, 2097174, F2, 25) (dual of [2097174, 2096920, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2253, 2097152, F2, 25) (dual of [2097152, 2096899, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2232, 2097152, F2, 23) (dual of [2097152, 2096920, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 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(24) ⊂ Ce(22) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2254, 2097174, F2, 25) (dual of [2097174, 2096920, 26]-code), using
- OOA 8-folding [i] based on linear OA(2256, 2097176, F2, 25) (dual of [2097176, 2096920, 26]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2256, 262147, F2, 8, 25) (dual of [(262147, 8), 2096920, 26]-NRT-code), using
(257−25, 257, large)-Net in Base 2 — Upper bound on s
There is no (232, 257, large)-net in base 2, because
- 23 times m-reduction [i] would yield (232, 234, large)-net in base 2, but