Best Known (110−9, 110, s)-Nets in Base 2
(110−9, 110, 2097407)-Net over F2 — Constructive and digital
Digital (101, 110, 2097407)-net over F2, using
- 21 times duplication [i] based on digital (100, 109, 2097407)-net over F2, using
- net defined by OOA [i] based on linear OOA(2109, 2097407, F2, 9, 9) (dual of [(2097407, 9), 18876554, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2109, 2097407, F2, 8, 9) (dual of [(2097407, 8), 16779147, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(216, 257, F2, 8, 4) (dual of [(257, 8), 2040, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(216, 257, F2, 4, 4) (dual of [(257, 4), 1012, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(216, 257, F2, 3, 4) (dual of [(257, 3), 755, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (12, 16, 257)-net over F2, using
- appending kth column [i] based on linear OOA(216, 257, F2, 3, 4) (dual of [(257, 3), 755, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(216, 257, F2, 4, 4) (dual of [(257, 4), 1012, 5]-NRT-code), using
- linear OOA(293, 2097150, F2, 8, 9) (dual of [(2097150, 8), 16777107, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(293, 8388601, F2, 9) (dual of [8388601, 8388508, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(293, large, F2, 9) (dual of [large, large−93, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(293, large, F2, 9) (dual of [large, large−93, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(293, 8388601, F2, 9) (dual of [8388601, 8388508, 10]-code), using
- linear OOA(216, 257, F2, 8, 4) (dual of [(257, 8), 2040, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(2109, 2097407, F2, 8, 9) (dual of [(2097407, 8), 16779147, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(2109, 2097407, F2, 9, 9) (dual of [(2097407, 9), 18876554, 10]-NRT-code), using
(110−9, 110, 2796466)-Net over F2 — Digital
Digital (101, 110, 2796466)-net over F2, using
- net defined by OOA [i] based on linear OOA(2110, 2796466, F2, 9, 9) (dual of [(2796466, 9), 25168084, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2110, 2796466, F2, 8, 9) (dual of [(2796466, 8), 22371618, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2110, 2796466, F2, 3, 9) (dual of [(2796466, 3), 8389288, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(217, 265, F2, 3, 4) (dual of [(265, 3), 778, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(217, 265, F2, 4) (dual of [265, 248, 5]-code), using
- 1 times truncation [i] based on linear OA(218, 266, F2, 5) (dual of [266, 248, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(217, 256, F2, 5) (dual of [256, 239, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(29, 256, F2, 3) (dual of [256, 247, 4]-code or 256-cap in PG(8,2)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(29, 10, F2, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,2)), using
- dual of repetition code with length 10 [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
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- 1 times truncation [i] based on linear OA(218, 266, F2, 5) (dual of [266, 248, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(217, 265, F2, 4) (dual of [265, 248, 5]-code), using
- linear OOA(293, 2796201, F2, 3, 9) (dual of [(2796201, 3), 8388510, 10]-NRT-code), using
- OOA 3-folding [i] based on linear OA(293, large, F2, 9) (dual of [large, large−93, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(293, large, F2, 9) (dual of [large, large−93, 10]-code), using
- linear OOA(217, 265, F2, 3, 4) (dual of [(265, 3), 778, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2110, 2796466, F2, 3, 9) (dual of [(2796466, 3), 8389288, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2110, 2796466, F2, 8, 9) (dual of [(2796466, 8), 22371618, 10]-NRT-code), using
(110−9, 110, large)-Net in Base 2 — Upper bound on s
There is no (101, 110, large)-net in base 2, because
- 7 times m-reduction [i] would yield (101, 103, large)-net in base 2, but