Best Known (217, 249, s)-Nets in Base 2
(217, 249, 2050)-Net over F2 — Constructive and digital
Digital (217, 249, 2050)-net over F2, using
- 22 times duplication [i] based on digital (215, 247, 2050)-net over F2, using
- t-expansion [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
- t-expansion [i] based on digital (214, 247, 2050)-net over F2, using
(217, 249, 6561)-Net over F2 — Digital
Digital (217, 249, 6561)-net over F2, using
- 22 times duplication [i] based on digital (215, 247, 6561)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2247, 6561, F2, 5, 32) (dual of [(6561, 5), 32558, 33]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2247, 32805, F2, 32) (dual of [32805, 32558, 33]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(2248, 32806, F2, 33) (dual of [32806, 32558, 34]-code), using
- OOA 5-folding [i] based on linear OA(2247, 32805, F2, 32) (dual of [32805, 32558, 33]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2247, 6561, F2, 5, 32) (dual of [(6561, 5), 32558, 33]-NRT-code), using
(217, 249, 329069)-Net in Base 2 — Upper bound on s
There is no (217, 249, 329070)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 904 668070 718771 793345 884977 979755 523296 499553 520135 172319 538644 164059 349998 > 2249 [i]