Best Known (213, 213+18, s)-Nets in Base 2
(213, 213+18, 932096)-Net over F2 — Constructive and digital
Digital (213, 231, 932096)-net over F2, using
- 1 times m-reduction [i] based on digital (213, 232, 932096)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (12, 21, 29)-net over F2, using
- digital (192, 211, 932067)-net over F2, using
- net defined by OOA [i] based on linear OOA(2211, 932067, F2, 21, 19) (dual of [(932067, 21), 19573196, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(2211, 2796202, F2, 3, 19) (dual of [(2796202, 3), 8388395, 20]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2208, 2796201, F2, 3, 19) (dual of [(2796201, 3), 8388395, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- OOA 3-folding [i] based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2208, 2796201, F2, 3, 19) (dual of [(2796201, 3), 8388395, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(2211, 2796202, F2, 3, 19) (dual of [(2796202, 3), 8388395, 20]-NRT-code), using
- net defined by OOA [i] based on linear OOA(2211, 932067, F2, 21, 19) (dual of [(932067, 21), 19573196, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
(213, 213+18, 1677750)-Net over F2 — Digital
Digital (213, 231, 1677750)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2231, 1677750, F2, 5, 18) (dual of [(1677750, 5), 8388519, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(224, 30, F2, 5, 9) (dual of [(30, 5), 126, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(224, 30, F2, 2, 9) (dual of [(30, 2), 36, 10]-NRT-code), using
- 21 times duplication [i] based on linear OOA(223, 30, F2, 2, 9) (dual of [(30, 2), 37, 10]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(221, 29, F2, 2, 9) (dual of [(29, 2), 37, 10]-NRT-code), using
- extracting embedded OOA [i] based on digital (12, 21, 29)-net over F2, using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(221, 29, F2, 2, 9) (dual of [(29, 2), 37, 10]-NRT-code), using
- 21 times duplication [i] based on linear OOA(223, 30, F2, 2, 9) (dual of [(30, 2), 37, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(224, 30, F2, 2, 9) (dual of [(30, 2), 36, 10]-NRT-code), using
- linear OOA(2207, 1677720, F2, 5, 18) (dual of [(1677720, 5), 8388393, 19]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- OOA 5-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-code), using
- linear OOA(224, 30, F2, 5, 9) (dual of [(30, 5), 126, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(213, 213+18, large)-Net in Base 2 — Upper bound on s
There is no (213, 231, large)-net in base 2, because
- 16 times m-reduction [i] would yield (213, 215, large)-net in base 2, but