Best Known (226, 226+22, s)-Nets in Base 2
(226, 226+22, 381303)-Net over F2 — Constructive and digital
Digital (226, 248, 381303)-net over F2, using
- net defined by OOA [i] based on linear OOA(2248, 381303, F2, 22, 22) (dual of [(381303, 22), 8388418, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(2248, 4194333, F2, 22) (dual of [4194333, 4194085, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2248, 4194335, F2, 22) (dual of [4194335, 4194087, 23]-code), using
- 1 times truncation [i] based on linear OA(2249, 4194336, F2, 23) (dual of [4194336, 4194087, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2243, 4194304, F2, 23) (dual of [4194304, 4194061, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2199, 4194304, F2, 19) (dual of [4194304, 4194105, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- 1 times truncation [i] based on linear OA(2249, 4194336, F2, 23) (dual of [4194336, 4194087, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2248, 4194335, F2, 22) (dual of [4194335, 4194087, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(2248, 4194333, F2, 22) (dual of [4194333, 4194085, 23]-code), using
(226, 226+22, 599190)-Net over F2 — Digital
Digital (226, 248, 599190)-net over F2, using
- 22 times duplication [i] based on digital (224, 246, 599190)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2246, 599190, F2, 7, 22) (dual of [(599190, 7), 4194084, 23]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2246, 4194330, F2, 22) (dual of [4194330, 4194084, 23]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2243, 4194327, F2, 22) (dual of [4194327, 4194084, 23]-code), using
- 1 times truncation [i] based on linear OA(2244, 4194328, F2, 23) (dual of [4194328, 4194084, 24]-code), using
- construction X4 applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2243, 4194304, F2, 23) (dual of [4194304, 4194061, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2221, 4194304, F2, 21) (dual of [4194304, 4194083, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(223, 24, F2, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,2)), using
- dual of repetition code with length 24 [i]
- linear OA(21, 24, F2, 1) (dual of [24, 23, 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 X4 applied to Ce(22) ⊂ Ce(20) [i] based on
- 1 times truncation [i] based on linear OA(2244, 4194328, F2, 23) (dual of [4194328, 4194084, 24]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2243, 4194327, F2, 22) (dual of [4194327, 4194084, 23]-code), using
- OOA 7-folding [i] based on linear OA(2246, 4194330, F2, 22) (dual of [4194330, 4194084, 23]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2246, 599190, F2, 7, 22) (dual of [(599190, 7), 4194084, 23]-NRT-code), using
(226, 226+22, large)-Net in Base 2 — Upper bound on s
There is no (226, 248, large)-net in base 2, because
- 20 times m-reduction [i] would yield (226, 228, large)-net in base 2, but