Best Known (38, 38+13, s)-Nets in Base 2
(38, 38+13, 75)-Net over F2 — Constructive and digital
Digital (38, 51, 75)-net over F2, using
- trace code for nets [i] based on digital (4, 17, 25)-net over F8, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 4 and N(F) ≥ 25, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
(38, 38+13, 122)-Net over F2 — Digital
Digital (38, 51, 122)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(251, 122, F2, 2, 13) (dual of [(122, 2), 193, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(251, 136, F2, 2, 13) (dual of [(136, 2), 221, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(251, 272, F2, 13) (dual of [272, 221, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(251, 273, F2, 13) (dual of [273, 222, 14]-code), using
- construction XX applied to C1 = C([253,8]), C2 = C([0,10]), C3 = C1 + C2 = C([0,8]), 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(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(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(233, 255, F2, 9) (dual of [255, 222, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [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([253,8]), C2 = C([0,10]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([253,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(251, 273, F2, 13) (dual of [273, 222, 14]-code), using
- OOA 2-folding [i] based on linear OA(251, 272, F2, 13) (dual of [272, 221, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(251, 136, F2, 2, 13) (dual of [(136, 2), 221, 14]-NRT-code), using
(38, 38+13, 957)-Net in Base 2 — Upper bound on s
There is no (38, 51, 958)-net in base 2, because
- 1 times m-reduction [i] would yield (38, 50, 958)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1131 856161 981092 > 250 [i]