Best Known (135−17, 135, s)-Nets in Base 2
(135−17, 135, 8196)-Net over F2 — Constructive and digital
Digital (118, 135, 8196)-net over F2, using
- net defined by OOA [i] based on linear OOA(2135, 8196, F2, 17, 17) (dual of [(8196, 17), 139197, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2135, 65569, F2, 17) (dual of [65569, 65434, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(2129, 65537, F2, 17) (dual of [65537, 65408, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(297, 65537, F2, 13) (dual of [65537, 65440, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [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 C([0,8]) ⊂ C([0,6]) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(2135, 65569, F2, 17) (dual of [65569, 65434, 18]-code), using
(135−17, 135, 13113)-Net over F2 — Digital
Digital (118, 135, 13113)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2135, 13113, F2, 5, 17) (dual of [(13113, 5), 65430, 18]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2135, 65565, F2, 17) (dual of [65565, 65430, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2135, 65569, F2, 17) (dual of [65569, 65434, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(2129, 65537, F2, 17) (dual of [65537, 65408, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(297, 65537, F2, 13) (dual of [65537, 65440, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [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 C([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2135, 65569, F2, 17) (dual of [65569, 65434, 18]-code), using
- OOA 5-folding [i] based on linear OA(2135, 65565, F2, 17) (dual of [65565, 65430, 18]-code), using
(135−17, 135, 414887)-Net in Base 2 — Upper bound on s
There is no (118, 135, 414888)-net in base 2, because
- 1 times m-reduction [i] would yield (118, 134, 414888)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 21778 239374 707365 997278 391104 319265 372852 > 2134 [i]