Best Known (213, 213+17, s)-Nets in Base 2
(213, 213+17, 1049090)-Net over F2 — Constructive and digital
Digital (213, 230, 1049090)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (37, 45, 515)-net over F2, using
- net defined by OOA [i] based on linear OOA(245, 515, F2, 8, 8) (dual of [(515, 8), 4075, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(245, 2060, F2, 8) (dual of [2060, 2015, 9]-code), using
- 1 times truncation [i] based on linear OA(246, 2061, F2, 9) (dual of [2061, 2015, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(245, 2048, F2, 9) (dual of [2048, 2003, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(234, 2048, F2, 7) (dual of [2048, 2014, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(212, 13, F2, 11) (dual of [13, 1, 12]-code), using
- strength reduction [i] based on linear OA(212, 13, F2, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,2)), using
- dual of repetition code with length 13 [i]
- strength reduction [i] based on linear OA(212, 13, F2, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,2)), using
- linear OA(21, 13, F2, 1) (dual of [13, 12, 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(246, 2061, F2, 9) (dual of [2061, 2015, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(245, 2060, F2, 8) (dual of [2060, 2015, 9]-code), using
- net defined by OOA [i] based on linear OOA(245, 515, F2, 8, 8) (dual of [(515, 8), 4075, 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 (37, 45, 515)-net over F2, using
(213, 213+17, 2098154)-Net over F2 — Digital
Digital (213, 230, 2098154)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2230, 2098154, F2, 4, 17) (dual of [(2098154, 4), 8392386, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(245, 1004, F2, 4, 8) (dual of [(1004, 4), 3971, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(245, 1004, F2, 2, 8) (dual of [(1004, 2), 1963, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(245, 1030, F2, 2, 8) (dual of [(1030, 2), 2015, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(245, 2060, F2, 8) (dual of [2060, 2015, 9]-code), using
- 1 times truncation [i] based on linear OA(246, 2061, F2, 9) (dual of [2061, 2015, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(245, 2048, F2, 9) (dual of [2048, 2003, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(234, 2048, F2, 7) (dual of [2048, 2014, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(212, 13, F2, 11) (dual of [13, 1, 12]-code), using
- strength reduction [i] based on linear OA(212, 13, F2, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,2)), using
- dual of repetition code with length 13 [i]
- strength reduction [i] based on linear OA(212, 13, F2, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,2)), using
- linear OA(21, 13, F2, 1) (dual of [13, 12, 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(246, 2061, F2, 9) (dual of [2061, 2015, 10]-code), using
- OOA 2-folding [i] based on linear OA(245, 2060, F2, 8) (dual of [2060, 2015, 9]-code), using
- discarding factors / shortening the dual code based on linear OOA(245, 1030, F2, 2, 8) (dual of [(1030, 2), 2015, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(245, 1004, F2, 2, 8) (dual of [(1004, 2), 1963, 9]-NRT-code), using
- linear OOA(2185, 2097150, F2, 4, 17) (dual of [(2097150, 4), 8388415, 18]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2185, 8388600, F2, 17) (dual of [8388600, 8388415, 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 4-folding [i] based on linear OA(2185, 8388600, F2, 17) (dual of [8388600, 8388415, 18]-code), using
- linear OOA(245, 1004, F2, 4, 8) (dual of [(1004, 4), 3971, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(213, 213+17, large)-Net in Base 2 — Upper bound on s
There is no (213, 230, large)-net in base 2, because
- 15 times m-reduction [i] would yield (213, 215, large)-net in base 2, but