Best Known (143, 143+17, s)-Nets in Base 2
(143, 143+17, 65541)-Net over F2 — Constructive and digital
Digital (143, 160, 65541)-net over F2, using
- net defined by OOA [i] based on linear OOA(2160, 65541, F2, 17, 17) (dual of [(65541, 17), 1114037, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2160, 524329, F2, 17) (dual of [524329, 524169, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2160, 524334, F2, 17) (dual of [524334, 524174, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(2153, 524289, F2, 17) (dual of [524289, 524136, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2115, 524289, F2, 13) (dual of [524289, 524174, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(27, 45, F2, 3) (dual of [45, 38, 4]-code or 45-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
- discarding factors / shortening the dual code based on linear OA(2160, 524334, F2, 17) (dual of [524334, 524174, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2160, 524329, F2, 17) (dual of [524329, 524169, 18]-code), using
(143, 143+17, 87389)-Net over F2 — Digital
Digital (143, 160, 87389)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2160, 87389, F2, 6, 17) (dual of [(87389, 6), 524174, 18]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2160, 524334, F2, 17) (dual of [524334, 524174, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(2153, 524289, F2, 17) (dual of [524289, 524136, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2115, 524289, F2, 13) (dual of [524289, 524174, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(27, 45, F2, 3) (dual of [45, 38, 4]-code or 45-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
- OOA 6-folding [i] based on linear OA(2160, 524334, F2, 17) (dual of [524334, 524174, 18]-code), using
(143, 143+17, 3619593)-Net in Base 2 — Upper bound on s
There is no (143, 160, 3619594)-net in base 2, because
- 1 times m-reduction [i] would yield (143, 159, 3619594)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 730750 955966 729623 907078 089984 564380 307790 416105 > 2159 [i]