Best Known (212, 244, s)-Nets in Base 2
(212, 244, 2049)-Net over F2 — Constructive and digital
Digital (212, 244, 2049)-net over F2, using
- 21 times duplication [i] based on digital (211, 243, 2049)-net over F2, using
- t-expansion [i] based on digital (210, 243, 2049)-net over F2, using
- net defined by OOA [i] based on linear OOA(2243, 2049, F2, 33, 33) (dual of [(2049, 33), 67374, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2243, 32785, F2, 33) (dual of [32785, 32542, 34]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2242, 32784, F2, 33) (dual of [32784, 32542, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- 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(2226, 32768, F2, 31) (dual of [32768, 32542, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(32) ⊂ Ce(30) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2242, 32784, F2, 33) (dual of [32784, 32542, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2243, 32785, F2, 33) (dual of [32785, 32542, 34]-code), using
- net defined by OOA [i] based on linear OOA(2243, 2049, F2, 33, 33) (dual of [(2049, 33), 67374, 34]-NRT-code), using
- t-expansion [i] based on digital (210, 243, 2049)-net over F2, using
(212, 244, 6134)-Net over F2 — Digital
Digital (212, 244, 6134)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2244, 6134, F2, 5, 32) (dual of [(6134, 5), 30426, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2244, 6557, F2, 5, 32) (dual of [(6557, 5), 32541, 33]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2243, 6557, F2, 5, 32) (dual of [(6557, 5), 32542, 33]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2243, 32785, F2, 32) (dual of [32785, 32542, 33]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2241, 32783, F2, 32) (dual of [32783, 32542, 33]-code), using
- 1 times truncation [i] based on linear OA(2242, 32784, F2, 33) (dual of [32784, 32542, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- 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(2226, 32768, F2, 31) (dual of [32768, 32542, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(32) ⊂ Ce(30) [i] based on
- 1 times truncation [i] based on linear OA(2242, 32784, F2, 33) (dual of [32784, 32542, 34]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2241, 32783, F2, 32) (dual of [32783, 32542, 33]-code), using
- OOA 5-folding [i] based on linear OA(2243, 32785, F2, 32) (dual of [32785, 32542, 33]-code), using
- 21 times duplication [i] based on linear OOA(2243, 6557, F2, 5, 32) (dual of [(6557, 5), 32542, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2244, 6557, F2, 5, 32) (dual of [(6557, 5), 32541, 33]-NRT-code), using
(212, 244, 264976)-Net in Base 2 — Upper bound on s
There is no (212, 244, 264977)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 28 270108 491647 506730 709356 468277 865962 506291 707426 986111 587294 095266 088209 > 2244 [i]