Best Known (234−29, 234, s)-Nets in Base 2
(234−29, 234, 4684)-Net over F2 — Constructive and digital
Digital (205, 234, 4684)-net over F2, using
- 21 times duplication [i] based on digital (204, 233, 4684)-net over F2, using
- net defined by OOA [i] based on linear OOA(2233, 4684, F2, 29, 29) (dual of [(4684, 29), 135603, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2233, 65577, F2, 29) (dual of [65577, 65344, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2232, 65576, F2, 29) (dual of [65576, 65344, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2225, 65537, F2, 29) (dual of [65537, 65312, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2193, 65537, F2, 25) (dual of [65537, 65344, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,14]) ⊂ C([0,12]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2232, 65576, F2, 29) (dual of [65576, 65344, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2233, 65577, F2, 29) (dual of [65577, 65344, 30]-code), using
- net defined by OOA [i] based on linear OOA(2233, 4684, F2, 29, 29) (dual of [(4684, 29), 135603, 30]-NRT-code), using
(234−29, 234, 10929)-Net over F2 — Digital
Digital (205, 234, 10929)-net over F2, using
- 22 times duplication [i] based on digital (203, 232, 10929)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2232, 10929, F2, 6, 29) (dual of [(10929, 6), 65342, 30]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2232, 65574, F2, 29) (dual of [65574, 65342, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2232, 65576, F2, 29) (dual of [65576, 65344, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2225, 65537, F2, 29) (dual of [65537, 65312, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2193, 65537, F2, 25) (dual of [65537, 65344, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2232, 65576, F2, 29) (dual of [65576, 65344, 30]-code), using
- OOA 6-folding [i] based on linear OA(2232, 65574, F2, 29) (dual of [65574, 65342, 30]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2232, 10929, F2, 6, 29) (dual of [(10929, 6), 65342, 30]-NRT-code), using
(234−29, 234, 618646)-Net in Base 2 — Upper bound on s
There is no (205, 234, 618647)-net in base 2, because
- 1 times m-reduction [i] would yield (205, 233, 618647)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 13803 647967 572988 299255 190413 448060 425548 587834 376753 290628 808140 947618 > 2233 [i]