Best Known (143−16, 143, s)-Nets in Base 2
(143−16, 143, 16389)-Net over F2 — Constructive and digital
Digital (127, 143, 16389)-net over F2, using
- net defined by OOA [i] based on linear OOA(2143, 16389, F2, 16, 16) (dual of [(16389, 16), 262081, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(2143, 131112, F2, 16) (dual of [131112, 130969, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2143, 131113, F2, 16) (dual of [131113, 130970, 17]-code), using
- 1 times truncation [i] based on linear OA(2144, 131114, F2, 17) (dual of [131114, 130970, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(2137, 131073, F2, 17) (dual of [131073, 130936, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2103, 131073, F2, 13) (dual of [131073, 130970, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(27, 41, F2, 3) (dual of [41, 34, 4]-code or 41-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(2144, 131114, F2, 17) (dual of [131114, 130970, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2143, 131113, F2, 16) (dual of [131113, 130970, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(2143, 131112, F2, 16) (dual of [131112, 130969, 17]-code), using
(143−16, 143, 31241)-Net over F2 — Digital
Digital (127, 143, 31241)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2143, 31241, F2, 4, 16) (dual of [(31241, 4), 124821, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2143, 32778, F2, 4, 16) (dual of [(32778, 4), 130969, 17]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2143, 131112, F2, 16) (dual of [131112, 130969, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2143, 131113, F2, 16) (dual of [131113, 130970, 17]-code), using
- 1 times truncation [i] based on linear OA(2144, 131114, F2, 17) (dual of [131114, 130970, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(2137, 131073, F2, 17) (dual of [131073, 130936, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2103, 131073, F2, 13) (dual of [131073, 130970, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(27, 41, F2, 3) (dual of [41, 34, 4]-code or 41-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(2144, 131114, F2, 17) (dual of [131114, 130970, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2143, 131113, F2, 16) (dual of [131113, 130970, 17]-code), using
- OOA 4-folding [i] based on linear OA(2143, 131112, F2, 16) (dual of [131112, 130969, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(2143, 32778, F2, 4, 16) (dual of [(32778, 4), 130969, 17]-NRT-code), using
(143−16, 143, 904889)-Net in Base 2 — Upper bound on s
There is no (127, 143, 904890)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 11 150387 010534 107866 655762 440468 571876 002819 > 2143 [i]