Best Known (65, 65+23, s)-Nets in Base 2
(65, 65+23, 84)-Net over F2 — Constructive and digital
Digital (65, 88, 84)-net over F2, using
- 2 times m-reduction [i] based on digital (65, 90, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 30, 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, 30, 28)-net over F8, using
(65, 65+23, 137)-Net over F2 — Digital
Digital (65, 88, 137)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(288, 137, F2, 2, 23) (dual of [(137, 2), 186, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(288, 274, F2, 23) (dual of [274, 186, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(288, 275, F2, 23) (dual of [275, 187, 24]-code), using
- adding a parity check bit [i] based on linear OA(287, 274, F2, 22) (dual of [274, 187, 23]-code), using
- construction XX applied to C1 = C([253,18]), C2 = C([1,20]), C3 = C1 + C2 = C([1,18]), 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(276, 255, F2, 20) (dual of [255, 179, 21]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(268, 255, F2, 18) (dual of [255, 187, 19]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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), 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 (see above)
- construction XX applied to C1 = C([253,18]), C2 = C([1,20]), C3 = C1 + C2 = C([1,18]), and C∩ = C1 ∩ C2 = C([253,20]) [i] based on
- adding a parity check bit [i] based on linear OA(287, 274, F2, 22) (dual of [274, 187, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(288, 275, F2, 23) (dual of [275, 187, 24]-code), using
- OOA 2-folding [i] based on linear OA(288, 274, F2, 23) (dual of [274, 186, 24]-code), using
(65, 65+23, 1164)-Net in Base 2 — Upper bound on s
There is no (65, 88, 1165)-net in base 2, because
- 1 times m-reduction [i] would yield (65, 87, 1165)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 156 027825 323353 031163 549728 > 287 [i]