Best Known (137, 137+21, s)-Nets in Base 2
(137, 137+21, 3280)-Net over F2 — Constructive and digital
Digital (137, 158, 3280)-net over F2, using
- 21 times duplication [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
(137, 137+21, 6561)-Net over F2 — Digital
Digital (137, 158, 6561)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2158, 6561, F2, 5, 21) (dual of [(6561, 5), 32647, 22]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2158, 32805, F2, 21) (dual of [32805, 32647, 22]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(2158, 32806, F2, 21) (dual of [32806, 32648, 22]-code), using
- OOA 5-folding [i] based on linear OA(2158, 32805, F2, 21) (dual of [32805, 32647, 22]-code), using
(137, 137+21, 241057)-Net in Base 2 — Upper bound on s
There is no (137, 158, 241058)-net in base 2, because
- 1 times m-reduction [i] would yield (137, 157, 241058)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 182689 537745 405053 617919 796626 940084 562887 998128 > 2157 [i]