Best Known (159−16, 159, s)-Nets in Base 2
(159−16, 159, 65541)-Net over F2 — Constructive and digital
Digital (143, 159, 65541)-net over F2, using
- net defined by OOA [i] based on linear OOA(2159, 65541, F2, 16, 16) (dual of [(65541, 16), 1048497, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(2159, 524328, F2, 16) (dual of [524328, 524169, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2159, 524333, F2, 16) (dual of [524333, 524174, 17]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2160, 524334, F2, 17) (dual of [524334, 524174, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2159, 524333, F2, 16) (dual of [524333, 524174, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(2159, 524328, F2, 16) (dual of [524328, 524169, 17]-code), using
(159−16, 159, 104866)-Net over F2 — Digital
Digital (143, 159, 104866)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2159, 104866, F2, 5, 16) (dual of [(104866, 5), 524171, 17]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2159, 524330, F2, 16) (dual of [524330, 524171, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2159, 524333, F2, 16) (dual of [524333, 524174, 17]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2160, 524334, F2, 17) (dual of [524334, 524174, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2159, 524333, F2, 16) (dual of [524333, 524174, 17]-code), using
- OOA 5-folding [i] based on linear OA(2159, 524330, F2, 16) (dual of [524330, 524171, 17]-code), using
(159−16, 159, 3619593)-Net in Base 2 — Upper bound on s
There is no (143, 159, 3619594)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 730750 955966 729623 907078 089984 564380 307790 416105 > 2159 [i]