Best Known (44, 59, s)-Nets in Base 2
(44, 59, 83)-Net over F2 — Constructive and digital
Digital (44, 59, 83)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (11, 18, 41)-net over F2, using
- digital (26, 41, 42)-net over F2, using
- 1 times m-reduction [i] based on digital (26, 42, 42)-net over F2, using
(44, 59, 126)-Net over F2 — Digital
Digital (44, 59, 126)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(259, 126, F2, 2, 15) (dual of [(126, 2), 193, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(259, 136, F2, 2, 15) (dual of [(136, 2), 213, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(259, 272, F2, 15) (dual of [272, 213, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(259, 273, F2, 15) (dual of [273, 214, 16]-code), using
- construction XX applied to C1 = C([253,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([253,12]) [i] based on
- linear OA(249, 255, F2, 13) (dual of [255, 206, 14]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,10}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(249, 255, F2, 13) (dual of [255, 206, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(257, 255, F2, 15) (dual of [255, 198, 16]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,12}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(241, 255, F2, 11) (dual of [255, 214, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(21, 9, F2, 1) (dual of [9, 8, 2]-code) (see above)
- construction XX applied to C1 = C([253,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([253,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(259, 273, F2, 15) (dual of [273, 214, 16]-code), using
- OOA 2-folding [i] based on linear OA(259, 272, F2, 15) (dual of [272, 213, 16]-code), using
- discarding factors / shortening the dual code based on linear OOA(259, 136, F2, 2, 15) (dual of [(136, 2), 213, 16]-NRT-code), using
(44, 59, 1044)-Net in Base 2 — Upper bound on s
There is no (44, 59, 1045)-net in base 2, because
- 1 times m-reduction [i] would yield (44, 58, 1045)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 288547 312986 043640 > 258 [i]