Best Known (44, 44+14, s)-Nets in Base 2
(44, 44+14, 84)-Net over F2 — Constructive and digital
Digital (44, 58, 84)-net over F2, using
- 21 times duplication [i] based on digital (43, 57, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 19, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- trace code for nets [i] based on digital (5, 19, 28)-net over F8, using
(44, 44+14, 136)-Net over F2 — Digital
Digital (44, 58, 136)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(258, 136, F2, 2, 14) (dual of [(136, 2), 214, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(258, 272, F2, 14) (dual of [272, 214, 15]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(259, 273, F2, 15) (dual of [273, 214, 16]-code), using
- OOA 2-folding [i] based on linear OA(258, 272, F2, 14) (dual of [272, 214, 15]-code), using
(44, 44+14, 1044)-Net in Base 2 — Upper bound on s
There is no (44, 58, 1045)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 288547 312986 043640 > 258 [i]