Best Known (182−27, 182, s)-Nets in Base 2
(182−27, 182, 633)-Net over F2 — Constructive and digital
Digital (155, 182, 633)-net over F2, using
- net defined by OOA [i] based on linear OOA(2182, 633, F2, 27, 27) (dual of [(633, 27), 16909, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2182, 8230, F2, 27) (dual of [8230, 8048, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2182, 8237, F2, 27) (dual of [8237, 8055, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- linear OA(2170, 8192, F2, 27) (dual of [8192, 8022, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2131, 8192, F2, 21) (dual of [8192, 8061, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, 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
- discarding factors / shortening the dual code based on linear OA(2182, 8237, F2, 27) (dual of [8237, 8055, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2182, 8230, F2, 27) (dual of [8230, 8048, 28]-code), using
(182−27, 182, 2060)-Net over F2 — Digital
Digital (155, 182, 2060)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2182, 2060, F2, 4, 27) (dual of [(2060, 4), 8058, 28]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2182, 8240, F2, 27) (dual of [8240, 8058, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- linear OA(2170, 8192, F2, 27) (dual of [8192, 8022, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2131, 8192, F2, 21) (dual of [8192, 8061, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, 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
- OOA 4-folding [i] based on linear OA(2182, 8240, F2, 27) (dual of [8240, 8058, 28]-code), using
(182−27, 182, 88018)-Net in Base 2 — Upper bound on s
There is no (155, 182, 88019)-net in base 2, because
- 1 times m-reduction [i] would yield (155, 181, 88019)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 065008 956278 286406 408259 938998 674052 635696 344789 565952 > 2181 [i]