Best Known (229, 229+25, s)-Nets in Base 2
(229, 229+25, 174764)-Net over F2 — Constructive and digital
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
(229, 229+25, 262146)-Net over F2 — Digital
Digital (229, 254, 262146)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2254, 262146, F2, 8, 25) (dual of [(262146, 8), 2096914, 26]-NRT-code), using
- OOA 8-folding [i] based on linear OA(2254, 2097168, F2, 25) (dual of [2097168, 2096914, 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 8-folding [i] based on linear OA(2254, 2097168, F2, 25) (dual of [2097168, 2096914, 26]-code), using
(229, 229+25, large)-Net in Base 2 — Upper bound on s
There is no (229, 254, large)-net in base 2, because
- 23 times m-reduction [i] would yield (229, 231, large)-net in base 2, but