Best Known (220, 252, s)-Nets in Base 2
(220, 252, 2050)-Net over F2 — Constructive and digital
Digital (220, 252, 2050)-net over F2, using
- 25 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
(220, 252, 6563)-Net over F2 — Digital
Digital (220, 252, 6563)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2252, 6563, F2, 5, 32) (dual of [(6563, 5), 32563, 33]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2252, 32815, F2, 32) (dual of [32815, 32563, 33]-code), using
- 1 times truncation [i] based on linear OA(2253, 32816, F2, 33) (dual of [32816, 32563, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(26) [i] based on
- linear OA(2241, 32768, F2, 33) (dual of [32768, 32527, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2196, 32768, F2, 27) (dual of [32768, 32572, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- extracting embedded orthogonal array [i] based on digital (7, 11, 47)-net over F2, using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- construction X applied to Ce(32) ⊂ Ce(26) [i] based on
- 1 times truncation [i] based on linear OA(2253, 32816, F2, 33) (dual of [32816, 32563, 34]-code), using
- OOA 5-folding [i] based on linear OA(2252, 32815, F2, 32) (dual of [32815, 32563, 33]-code), using
(220, 252, 374743)-Net in Base 2 — Upper bound on s
There is no (220, 252, 374744)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 7237 209425 172810 835726 150083 516925 669564 098550 038456 729380 207850 980427 156105 > 2252 [i]