Best Known (234, 234+17, s)-Nets in Base 2
(234, 234+17, 1064963)-Net over F2 — Constructive and digital
Digital (234, 251, 1064963)-net over F2, using
- 21 times duplication [i] based on digital (233, 250, 1064963)-net over F2, using
- (u, u+v)-construction [i] based on
- 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
- net defined by OOA [i] based on linear OOA(265, 16388, F2, 8, 8) (dual of [(16388, 8), 131039, 9]-NRT-code), using
- digital (168, 185, 1048575)-net over F2, using
- net defined by OOA [i] based on linear OOA(2185, 1048575, F2, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2185, 8388601, F2, 17) (dual of [8388601, 8388416, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2185, 8388601, F2, 17) (dual of [8388601, 8388416, 18]-code), using
- net defined by OOA [i] based on linear OOA(2185, 1048575, F2, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
- digital (57, 65, 16388)-net over F2, using
- (u, u+v)-construction [i] based on
(234, 234+17, 2818052)-Net over F2 — Digital
Digital (234, 251, 2818052)-net over F2, using
- 21 times duplication [i] based on digital (233, 250, 2818052)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2250, 2818052, F2, 3, 17) (dual of [(2818052, 3), 8453906, 18]-NRT-code), using
- (u, u+v)-construction [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
- linear OOA(2185, 2796201, F2, 3, 17) (dual of [(2796201, 3), 8388418, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- linear OOA(265, 21851, F2, 3, 8) (dual of [(21851, 3), 65488, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2250, 2818052, F2, 3, 17) (dual of [(2818052, 3), 8453906, 18]-NRT-code), using
(234, 234+17, large)-Net in Base 2 — Upper bound on s
There is no (234, 251, large)-net in base 2, because
- 15 times m-reduction [i] would yield (234, 236, large)-net in base 2, but