Best Known (127−21, 127, s)-Nets in Base 2
(127−21, 127, 412)-Net over F2 — Constructive and digital
Digital (106, 127, 412)-net over F2, using
- net defined by OOA [i] based on linear OOA(2127, 412, F2, 21, 21) (dual of [(412, 21), 8525, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2127, 4121, F2, 21) (dual of [4121, 3994, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2127, 4127, F2, 21) (dual of [4127, 4000, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2121, 4097, F2, 21) (dual of [4097, 3976, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(297, 4097, F2, 17) (dual of [4097, 4000, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2127, 4127, F2, 21) (dual of [4127, 4000, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2127, 4121, F2, 21) (dual of [4121, 3994, 22]-code), using
(127−21, 127, 1102)-Net over F2 — Digital
Digital (106, 127, 1102)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2127, 1102, F2, 3, 21) (dual of [(1102, 3), 3179, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2127, 1375, F2, 3, 21) (dual of [(1375, 3), 3998, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2127, 4125, F2, 21) (dual of [4125, 3998, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2127, 4127, F2, 21) (dual of [4127, 4000, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2121, 4097, F2, 21) (dual of [4097, 3976, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(297, 4097, F2, 17) (dual of [4097, 4000, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2127, 4127, F2, 21) (dual of [4127, 4000, 22]-code), using
- OOA 3-folding [i] based on linear OA(2127, 4125, F2, 21) (dual of [4125, 3998, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(2127, 1375, F2, 3, 21) (dual of [(1375, 3), 3998, 22]-NRT-code), using
(127−21, 127, 28101)-Net in Base 2 — Upper bound on s
There is no (106, 127, 28102)-net in base 2, because
- 1 times m-reduction [i] would yield (106, 126, 28102)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 85 084221 310332 621232 856161 772528 472464 > 2126 [i]