Best Known (41, 41+12, s)-Nets in Base 2
(41, 41+12, 96)-Net over F2 — Constructive and digital
Digital (41, 53, 96)-net over F2, using
- 21 times duplication [i] based on digital (40, 52, 96)-net over F2, using
- trace code for nets [i] based on digital (1, 13, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- trace code for nets [i] based on digital (1, 13, 24)-net over F16, using
(41, 41+12, 154)-Net over F2 — Digital
Digital (41, 53, 154)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(253, 154, F2, 12) (dual of [154, 101, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(253, 276, F2, 12) (dual of [276, 223, 13]-code), using
- 1 times truncation [i] based on linear OA(254, 277, F2, 13) (dual of [277, 223, 14]-code), using
- construction XX applied to C1 = C([253,8]), C2 = C([1,10]), C3 = C1 + C2 = C([1,8]), and C∩ = C1 ∩ C2 = C([253,10]) [i] based on
- linear OA(241, 255, F2, 11) (dual of [255, 214, 12]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(240, 255, F2, 10) (dual of [255, 215, 11]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(232, 255, F2, 8) (dual of [255, 223, 9]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(24, 13, F2, 2) (dual of [13, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(24, 15, F2, 2) (dual of [15, 11, 3]-code), using
- Hamming code H(4,2) [i]
- discarding factors / shortening the dual code based on linear OA(24, 15, F2, 2) (dual of [15, 11, 3]-code), using
- 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
- construction XX applied to C1 = C([253,8]), C2 = C([1,10]), C3 = C1 + C2 = C([1,8]), and C∩ = C1 ∩ C2 = C([253,10]) [i] based on
- 1 times truncation [i] based on linear OA(254, 277, F2, 13) (dual of [277, 223, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(253, 276, F2, 12) (dual of [276, 223, 13]-code), using
(41, 41+12, 1357)-Net in Base 2 — Upper bound on s
There is no (41, 53, 1358)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 9042 354763 474012 > 253 [i]