Best Known (139, 139+20, s)-Nets in Base 2
(139, 139+20, 3280)-Net over F2 — Constructive and digital
Digital (139, 159, 3280)-net over F2, using
- 22 times duplication [i] based on digital (137, 157, 3280)-net over F2, using
- t-expansion [i] based on digital (136, 157, 3280)-net over F2, using
- net defined by OOA [i] based on linear OOA(2157, 3280, F2, 21, 21) (dual of [(3280, 21), 68723, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2157, 32801, F2, 21) (dual of [32801, 32644, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2151, 32769, F2, 21) (dual of [32769, 32618, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2121, 32769, F2, 17) (dual of [32769, 32648, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,10]) ⊂ C([0,8]) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(2157, 32801, F2, 21) (dual of [32801, 32644, 22]-code), using
- net defined by OOA [i] based on linear OOA(2157, 3280, F2, 21, 21) (dual of [(3280, 21), 68723, 22]-NRT-code), using
- t-expansion [i] based on digital (136, 157, 3280)-net over F2, using
(139, 139+20, 8201)-Net over F2 — Digital
Digital (139, 159, 8201)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2159, 8201, F2, 4, 20) (dual of [(8201, 4), 32645, 21]-NRT-code), using
- 1 step truncation [i] based on linear OOA(2160, 8202, F2, 4, 21) (dual of [(8202, 4), 32648, 22]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2160, 32808, F2, 21) (dual of [32808, 32648, 22]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2158, 32806, F2, 21) (dual of [32806, 32648, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2151, 32769, F2, 21) (dual of [32769, 32618, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2121, 32769, F2, 17) (dual of [32769, 32648, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(27, 37, F2, 3) (dual of [37, 30, 4]-code or 37-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([0,10]) ⊂ C([0,8]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2158, 32806, F2, 21) (dual of [32806, 32648, 22]-code), using
- OOA 4-folding [i] based on linear OA(2160, 32808, F2, 21) (dual of [32808, 32648, 22]-code), using
- 1 step truncation [i] based on linear OOA(2160, 8202, F2, 4, 21) (dual of [(8202, 4), 32648, 22]-NRT-code), using
(139, 139+20, 276904)-Net in Base 2 — Upper bound on s
There is no (139, 159, 276905)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 730755 932306 816381 509067 941078 454238 464817 835284 > 2159 [i]