Best Known (48, 62, s)-Nets in Base 2
(48, 62, 96)-Net over F2 — Constructive and digital
Digital (48, 62, 96)-net over F2, using
- 22 times duplication [i] based on digital (46, 60, 96)-net over F2, using
- trace code for nets [i] based on digital (1, 15, 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, 15, 24)-net over F16, using
(48, 62, 165)-Net over F2 — Digital
Digital (48, 62, 165)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(262, 165, F2, 14) (dual of [165, 103, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(262, 279, F2, 14) (dual of [279, 217, 15]-code), using
- 1 times truncation [i] based on linear OA(263, 280, F2, 15) (dual of [280, 217, 16]-code), using
- construction XX applied to C1 = C([251,8]), C2 = C([0,10]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([251,10]) [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 = {−4,−3,…,8}, and designed minimum distance d ≥ |I|+1 = 14 [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(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 = {−4,−3,…,10}, and designed minimum distance d ≥ |I|+1 = 16 [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(25, 16, F2, 3) (dual of [16, 11, 4]-code or 16-cap in PG(4,2)), 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([251,8]), C2 = C([0,10]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([251,10]) [i] based on
- 1 times truncation [i] based on linear OA(263, 280, F2, 15) (dual of [280, 217, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(262, 279, F2, 14) (dual of [279, 217, 15]-code), using
(48, 62, 1557)-Net in Base 2 — Upper bound on s
There is no (48, 62, 1558)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 4 623146 224417 762244 > 262 [i]