Best Known (226, 253, s)-Nets in Base 2
(226, 253, 40331)-Net over F2 — Constructive and digital
Digital (226, 253, 40331)-net over F2, using
- 24 times duplication [i] based on digital (222, 249, 40331)-net over F2, using
- net defined by OOA [i] based on linear OOA(2249, 40331, F2, 27, 27) (dual of [(40331, 27), 1088688, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2249, 524304, F2, 27) (dual of [524304, 524055, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2249, 524308, F2, 27) (dual of [524308, 524059, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2248, 524288, F2, 27) (dual of [524288, 524040, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2229, 524288, F2, 25) (dual of [524288, 524059, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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(26) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2249, 524308, F2, 27) (dual of [524308, 524059, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2249, 524304, F2, 27) (dual of [524304, 524055, 28]-code), using
- net defined by OOA [i] based on linear OOA(2249, 40331, F2, 27, 27) (dual of [(40331, 27), 1088688, 28]-NRT-code), using
(226, 253, 65539)-Net over F2 — Digital
Digital (226, 253, 65539)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2253, 65539, F2, 8, 27) (dual of [(65539, 8), 524059, 28]-NRT-code), using
- OOA 8-folding [i] based on linear OA(2253, 524312, F2, 27) (dual of [524312, 524059, 28]-code), using
- 4 times code embedding in larger space [i] based on linear OA(2249, 524308, F2, 27) (dual of [524308, 524059, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2248, 524288, F2, 27) (dual of [524288, 524040, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2229, 524288, F2, 25) (dual of [524288, 524059, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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(26) ⊂ Ce(24) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(2249, 524308, F2, 27) (dual of [524308, 524059, 28]-code), using
- OOA 8-folding [i] based on linear OA(2253, 524312, F2, 27) (dual of [524312, 524059, 28]-code), using
(226, 253, 3879312)-Net in Base 2 — Upper bound on s
There is no (226, 253, 3879313)-net in base 2, because
- 1 times m-reduction [i] would yield (226, 252, 3879313)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 7237 024099 787629 072490 882075 299681 189231 797494 554580 856554 938340 478326 331784 > 2252 [i]