Best Known (223−31, 223, s)-Nets in Base 2
(223−31, 223, 1095)-Net over F2 — Constructive and digital
Digital (192, 223, 1095)-net over F2, using
- net defined by OOA [i] based on linear OOA(2223, 1095, F2, 31, 31) (dual of [(1095, 31), 33722, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2223, 16426, F2, 31) (dual of [16426, 16203, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2223, 16429, F2, 31) (dual of [16429, 16206, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2169, 16384, F2, 25) (dual of [16384, 16215, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, 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(2223, 16429, F2, 31) (dual of [16429, 16206, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2223, 16426, F2, 31) (dual of [16426, 16203, 32]-code), using
(223−31, 223, 3584)-Net over F2 — Digital
Digital (192, 223, 3584)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2223, 3584, F2, 4, 31) (dual of [(3584, 4), 14113, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2223, 4108, F2, 4, 31) (dual of [(4108, 4), 16209, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2223, 16432, F2, 31) (dual of [16432, 16209, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2169, 16384, F2, 25) (dual of [16384, 16215, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(30) ⊂ Ce(24) [i] based on
- OOA 4-folding [i] based on linear OA(2223, 16432, F2, 31) (dual of [16432, 16209, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(2223, 4108, F2, 4, 31) (dual of [(4108, 4), 16209, 32]-NRT-code), using
(223−31, 223, 183213)-Net in Base 2 — Upper bound on s
There is no (192, 223, 183214)-net in base 2, because
- 1 times m-reduction [i] would yield (192, 222, 183214)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6 740038 638230 713291 901976 926758 404878 514915 901025 885945 285988 394588 > 2222 [i]