Best Known (248−33, 248, s)-Nets in Base 2
(248−33, 248, 2050)-Net over F2 — Constructive and digital
Digital (215, 248, 2050)-net over F2, using
- 21 times duplication [i] based on digital (214, 247, 2050)-net over F2, using
- net defined by OOA [i] based on linear OOA(2247, 2050, F2, 33, 33) (dual of [(2050, 33), 67403, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2247, 32801, F2, 33) (dual of [32801, 32554, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(2241, 32769, F2, 33) (dual of [32769, 32528, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(2211, 32769, F2, 29) (dual of [32769, 32558, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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([0,16]) ⊂ C([0,14]) [i] based on
- OOA 16-folding and stacking with additional row [i] based on linear OA(2247, 32801, F2, 33) (dual of [32801, 32554, 34]-code), using
- net defined by OOA [i] based on linear OOA(2247, 2050, F2, 33, 33) (dual of [(2050, 33), 67403, 34]-NRT-code), using
(248−33, 248, 5554)-Net over F2 — Digital
Digital (215, 248, 5554)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2248, 5554, F2, 5, 33) (dual of [(5554, 5), 27522, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2248, 6561, F2, 5, 33) (dual of [(6561, 5), 32557, 34]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2248, 32805, F2, 33) (dual of [32805, 32557, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2248, 32806, F2, 33) (dual of [32806, 32558, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(2241, 32769, F2, 33) (dual of [32769, 32528, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(2211, 32769, F2, 29) (dual of [32769, 32558, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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([0,16]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2248, 32806, F2, 33) (dual of [32806, 32558, 34]-code), using
- OOA 5-folding [i] based on linear OA(2248, 32805, F2, 33) (dual of [32805, 32557, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(2248, 6561, F2, 5, 33) (dual of [(6561, 5), 32557, 34]-NRT-code), using
(248−33, 248, 301755)-Net in Base 2 — Upper bound on s
There is no (215, 248, 301756)-net in base 2, because
- 1 times m-reduction [i] would yield (215, 247, 301756)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 226 160180 162664 700821 527185 073848 519981 999527 252398 635871 302272 417385 176665 > 2247 [i]