Best Known (62, 62+21, s)-Nets in Base 2
(62, 62+21, 84)-Net over F2 — Constructive and digital
Digital (62, 83, 84)-net over F2, using
- t-expansion [i] based on digital (61, 83, 84)-net over F2, using
- 1 times m-reduction [i] based on digital (61, 84, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 28, 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, 28, 28)-net over F8, using
- 1 times m-reduction [i] based on digital (61, 84, 84)-net over F2, using
(62, 62+21, 140)-Net over F2 — Digital
Digital (62, 83, 140)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(283, 140, F2, 2, 21) (dual of [(140, 2), 197, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(283, 280, F2, 21) (dual of [280, 197, 22]-code), using
- construction XX applied to C1 = C([237,0]), C2 = C([241,2]), C3 = C1 + C2 = C([241,0]), and C∩ = C1 ∩ C2 = C([237,2]) [i] based on
- linear OA(269, 255, F2, 19) (dual of [255, 186, 20]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−18,−17,…,0}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(265, 255, F2, 17) (dual of [255, 190, 18]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−14,−13,…,2}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(277, 255, F2, 21) (dual of [255, 178, 22]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−18,−17,…,2}, and designed minimum distance d ≥ |I|+1 = 22 [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 = {−14,−13,…,0}, and designed minimum distance d ≥ |I|+1 = 16 [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([237,0]), C2 = C([241,2]), C3 = C1 + C2 = C([241,0]), and C∩ = C1 ∩ C2 = C([237,2]) [i] based on
- OOA 2-folding [i] based on linear OA(283, 280, F2, 21) (dual of [280, 197, 22]-code), using
(62, 62+21, 1317)-Net in Base 2 — Upper bound on s
There is no (62, 83, 1318)-net in base 2, because
- 1 times m-reduction [i] would yield (62, 82, 1318)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 4 860491 738935 604348 057144 > 282 [i]