Best Known (181, 181+27, s)-Nets in Base 2
(181, 181+27, 2524)-Net over F2 — Constructive and digital
Digital (181, 208, 2524)-net over F2, using
- net defined by OOA [i] based on linear OOA(2208, 2524, F2, 27, 27) (dual of [(2524, 27), 67940, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2208, 32813, F2, 27) (dual of [32813, 32605, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- 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(2151, 32768, F2, 21) (dual of [32768, 32617, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(212, 45, F2, 5) (dual of [45, 33, 6]-code), using
- discarding factors / shortening the dual code based on 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
- discarding factors / shortening the dual code based on linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- OOA 13-folding and stacking with additional row [i] based on linear OA(2208, 32813, F2, 27) (dual of [32813, 32605, 28]-code), using
(181, 181+27, 6563)-Net over F2 — Digital
Digital (181, 208, 6563)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2208, 6563, F2, 5, 27) (dual of [(6563, 5), 32607, 28]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2208, 32815, F2, 27) (dual of [32815, 32607, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, 32816, F2, 27) (dual of [32816, 32608, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- 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(2151, 32768, F2, 21) (dual of [32768, 32617, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(26) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2208, 32816, F2, 27) (dual of [32816, 32608, 28]-code), using
- OOA 5-folding [i] based on linear OA(2208, 32815, F2, 27) (dual of [32815, 32607, 28]-code), using
(181, 181+27, 352132)-Net in Base 2 — Upper bound on s
There is no (181, 208, 352133)-net in base 2, because
- 1 times m-reduction [i] would yield (181, 207, 352133)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 205 689321 918824 947450 937854 898763 327393 406323 266657 648376 317664 > 2207 [i]