Best Known (87−23, 87, s)-Nets in Base 2
(87−23, 87, 84)-Net over F2 — Constructive and digital
Digital (64, 87, 84)-net over F2, using
- t-expansion [i] based on digital (63, 87, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 29, 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, 29, 28)-net over F8, using
(87−23, 87, 132)-Net over F2 — Digital
Digital (64, 87, 132)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(287, 132, F2, 2, 23) (dual of [(132, 2), 177, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(287, 136, F2, 2, 23) (dual of [(136, 2), 185, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(287, 272, F2, 23) (dual of [272, 185, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(287, 273, F2, 23) (dual of [273, 186, 24]-code), using
- construction XX applied to C1 = C([253,18]), C2 = C([0,20]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([253,20]) [i] based on
- 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 = {−2,−1,…,18}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(277, 255, F2, 21) (dual of [255, 178, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(285, 255, F2, 23) (dual of [255, 170, 24]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,20}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(269, 255, F2, 19) (dual of [255, 186, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [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([253,18]), C2 = C([0,20]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([253,20]) [i] based on
- discarding factors / shortening the dual code based on linear OA(287, 273, F2, 23) (dual of [273, 186, 24]-code), using
- OOA 2-folding [i] based on linear OA(287, 272, F2, 23) (dual of [272, 185, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(287, 136, F2, 2, 23) (dual of [(136, 2), 185, 24]-NRT-code), using
(87−23, 87, 1092)-Net in Base 2 — Upper bound on s
There is no (64, 87, 1093)-net in base 2, because
- 1 times m-reduction [i] would yield (64, 86, 1093)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 78 099446 662108 013644 574976 > 286 [i]