Best Known (222, 222+26, s)-Nets in Base 2
(222, 222+26, 40331)-Net over F2 — Constructive and digital
Digital (222, 248, 40331)-net over F2, using
- net defined by OOA [i] based on linear OOA(2248, 40331, F2, 26, 26) (dual of [(40331, 26), 1048358, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(2248, 524303, F2, 26) (dual of [524303, 524055, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2248, 524307, F2, 26) (dual of [524307, 524059, 27]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2249, 524308, F2, 27) (dual of [524308, 524059, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2248, 524307, F2, 26) (dual of [524307, 524059, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(2248, 524303, F2, 26) (dual of [524303, 524055, 27]-code), using
(222, 222+26, 74901)-Net over F2 — Digital
Digital (222, 248, 74901)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2248, 74901, F2, 7, 26) (dual of [(74901, 7), 524059, 27]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2248, 524307, F2, 26) (dual of [524307, 524059, 27]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2249, 524308, F2, 27) (dual of [524308, 524059, 28]-code), using
- OOA 7-folding [i] based on linear OA(2248, 524307, F2, 26) (dual of [524307, 524059, 27]-code), using
(222, 222+26, 3134221)-Net in Base 2 — Upper bound on s
There is no (222, 248, 3134222)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 452 314718 879820 312919 321061 012410 655940 311472 074587 878382 293059 761211 393235 > 2248 [i]