Best Known (169−21, 169, s)-Nets in Base 2
(169−21, 169, 6557)-Net over F2 — Constructive and digital
Digital (148, 169, 6557)-net over F2, using
- 21 times duplication [i] based on digital (147, 168, 6557)-net over F2, using
- net defined by OOA [i] based on linear OOA(2168, 6557, F2, 21, 21) (dual of [(6557, 21), 137529, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2168, 65571, F2, 21) (dual of [65571, 65403, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2168, 65576, F2, 21) (dual of [65576, 65408, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2161, 65537, F2, 21) (dual of [65537, 65376, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2129, 65537, F2, 17) (dual of [65537, 65408, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(27, 39, F2, 3) (dual of [39, 32, 4]-code or 39-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(2168, 65576, F2, 21) (dual of [65576, 65408, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2168, 65571, F2, 21) (dual of [65571, 65403, 22]-code), using
- net defined by OOA [i] based on linear OOA(2168, 6557, F2, 21, 21) (dual of [(6557, 21), 137529, 22]-NRT-code), using
(169−21, 169, 12540)-Net over F2 — Digital
Digital (148, 169, 12540)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2169, 12540, F2, 5, 21) (dual of [(12540, 5), 62531, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2169, 13115, F2, 5, 21) (dual of [(13115, 5), 65406, 22]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2168, 13115, F2, 5, 21) (dual of [(13115, 5), 65407, 22]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2168, 65575, F2, 21) (dual of [65575, 65407, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2168, 65576, F2, 21) (dual of [65576, 65408, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2161, 65537, F2, 21) (dual of [65537, 65376, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2129, 65537, F2, 17) (dual of [65537, 65408, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(27, 39, F2, 3) (dual of [39, 32, 4]-code or 39-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(2168, 65576, F2, 21) (dual of [65576, 65408, 22]-code), using
- OOA 5-folding [i] based on linear OA(2168, 65575, F2, 21) (dual of [65575, 65407, 22]-code), using
- 21 times duplication [i] based on linear OOA(2168, 13115, F2, 5, 21) (dual of [(13115, 5), 65407, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2169, 13115, F2, 5, 21) (dual of [(13115, 5), 65406, 22]-NRT-code), using
(169−21, 169, 516735)-Net in Base 2 — Upper bound on s
There is no (148, 169, 516736)-net in base 2, because
- 1 times m-reduction [i] would yield (148, 168, 516736)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 374 150233 639276 719882 133100 091763 446366 659771 327313 > 2168 [i]