Best Known (119, 135, s)-Nets in Base 2
(119, 135, 8196)-Net over F2 — Constructive and digital
Digital (119, 135, 8196)-net over F2, using
- t-expansion [i] based on 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
- net defined by OOA [i] based on linear OOA(2135, 8196, F2, 17, 17) (dual of [(8196, 17), 139197, 18]-NRT-code), using
(119, 135, 16393)-Net over F2 — Digital
Digital (119, 135, 16393)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2135, 16393, F2, 4, 16) (dual of [(16393, 4), 65437, 17]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2135, 65572, F2, 16) (dual of [65572, 65437, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2135, 65575, F2, 16) (dual of [65575, 65440, 17]-code), using
- 1 times truncation [i] based on linear OA(2136, 65576, F2, 17) (dual of [65576, 65440, 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(27, 39, F2, 3) (dual of [39, 32, 4]-code or 39-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- 1 times truncation [i] based on linear OA(2136, 65576, F2, 17) (dual of [65576, 65440, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2135, 65575, F2, 16) (dual of [65575, 65440, 17]-code), using
- OOA 4-folding [i] based on linear OA(2135, 65572, F2, 16) (dual of [65572, 65437, 17]-code), using
(119, 135, 452439)-Net in Base 2 — Upper bound on s
There is no (119, 135, 452440)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 43556 776792 039013 807083 345983 275911 930354 > 2135 [i]