Best Known (253−31, 253, s)-Nets in Base 2
(253−31, 253, 4372)-Net over F2 — Constructive and digital
Digital (222, 253, 4372)-net over F2, using
- net defined by OOA [i] based on linear OOA(2253, 4372, F2, 31, 31) (dual of [(4372, 31), 135279, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2253, 65581, F2, 31) (dual of [65581, 65328, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(2241, 65536, F2, 31) (dual of [65536, 65295, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2193, 65536, F2, 25) (dual of [65536, 65343, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(30) ⊂ Ce(24) [i] based on
- OOA 15-folding and stacking with additional row [i] based on linear OA(2253, 65581, F2, 31) (dual of [65581, 65328, 32]-code), using
(253−31, 253, 10930)-Net over F2 — Digital
Digital (222, 253, 10930)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2253, 10930, F2, 6, 31) (dual of [(10930, 6), 65327, 32]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2253, 65580, F2, 31) (dual of [65580, 65327, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2253, 65581, F2, 31) (dual of [65581, 65328, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(2241, 65536, F2, 31) (dual of [65536, 65295, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2193, 65536, F2, 25) (dual of [65536, 65343, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(30) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2253, 65581, F2, 31) (dual of [65581, 65328, 32]-code), using
- OOA 6-folding [i] based on linear OA(2253, 65580, F2, 31) (dual of [65580, 65327, 32]-code), using
(253−31, 253, 732921)-Net in Base 2 — Upper bound on s
There is no (222, 253, 732922)-net in base 2, because
- 1 times m-reduction [i] would yield (222, 252, 732922)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 7237 062046 022686 181206 476785 152911 524493 870304 105952 639196 331926 625267 777256 > 2252 [i]