Best Known (57, 65, s)-Nets in Base 2
(57, 65, 16388)-Net over F2 — Constructive and digital
Digital (57, 65, 16388)-net over F2, using
- net defined by OOA [i] based on linear OOA(265, 16388, F2, 8, 8) (dual of [(16388, 8), 131039, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(265, 65552, F2, 8) (dual of [65552, 65487, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(265, 65553, F2, 8) (dual of [65553, 65488, 9]-code), using
- 1 times truncation [i] based on linear OA(266, 65554, F2, 9) (dual of [65554, 65488, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(265, 65536, F2, 9) (dual of [65536, 65471, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(249, 65536, F2, 7) (dual of [65536, 65487, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(217, 18, F2, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,2)), using
- dual of repetition code with length 18 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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 X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 1 times truncation [i] based on linear OA(266, 65554, F2, 9) (dual of [65554, 65488, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(265, 65553, F2, 8) (dual of [65553, 65488, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(265, 65552, F2, 8) (dual of [65552, 65487, 9]-code), using
(57, 65, 21851)-Net over F2 — Digital
Digital (57, 65, 21851)-net over F2, using
- net defined by OOA [i] based on linear OOA(265, 21851, F2, 8, 8) (dual of [(21851, 8), 174743, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(265, 21851, F2, 7, 8) (dual of [(21851, 7), 152892, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(265, 21851, F2, 3, 8) (dual of [(21851, 3), 65488, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(265, 65553, F2, 8) (dual of [65553, 65488, 9]-code), using
- 1 times truncation [i] based on linear OA(266, 65554, F2, 9) (dual of [65554, 65488, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(265, 65536, F2, 9) (dual of [65536, 65471, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(249, 65536, F2, 7) (dual of [65536, 65487, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(217, 18, F2, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,2)), using
- dual of repetition code with length 18 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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 X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 1 times truncation [i] based on linear OA(266, 65554, F2, 9) (dual of [65554, 65488, 10]-code), using
- OOA 3-folding [i] based on linear OA(265, 65553, F2, 8) (dual of [65553, 65488, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(265, 21851, F2, 3, 8) (dual of [(21851, 3), 65488, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(265, 21851, F2, 7, 8) (dual of [(21851, 7), 152892, 9]-NRT-code), using
(57, 65, 172494)-Net in Base 2 — Upper bound on s
There is no (57, 65, 172495)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 36 893528 271279 236881 > 265 [i]