Best Known (223, 241, s)-Nets in Base 2
(223, 241, 932133)-Net over F2 — Constructive and digital
Digital (223, 241, 932133)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (25, 34, 66)-net over F2, using
- net defined by OOA [i] based on linear OOA(234, 66, F2, 9, 9) (dual of [(66, 9), 560, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(234, 66, F2, 8, 9) (dual of [(66, 8), 494, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(234, 265, F2, 9) (dual of [265, 231, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(233, 256, F2, 9) (dual of [256, 223, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(225, 256, F2, 7) (dual of [256, 231, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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 X applied to Ce(8) ⊂ Ce(6) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(234, 265, F2, 9) (dual of [265, 231, 10]-code), using
- appending kth column [i] based on linear OOA(234, 66, F2, 8, 9) (dual of [(66, 8), 494, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(234, 66, F2, 9, 9) (dual of [(66, 9), 560, 10]-NRT-code), using
- digital (189, 207, 932067)-net over F2, using
- net defined by OOA [i] based on linear OOA(2207, 932067, F2, 18, 18) (dual of [(932067, 18), 16776999, 19]-NRT-code), using
- OA 9-folding and stacking [i] 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]
- OA 9-folding and stacking [i] based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- net defined by OOA [i] based on linear OOA(2207, 932067, F2, 18, 18) (dual of [(932067, 18), 16776999, 19]-NRT-code), using
- digital (25, 34, 66)-net over F2, using
(223, 241, 1743452)-Net over F2 — Digital
Digital (223, 241, 1743452)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2241, 1743452, F2, 4, 18) (dual of [(1743452, 4), 6973567, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2241, 2097217, F2, 4, 18) (dual of [(2097217, 4), 8388627, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OOA(2241, 4194434, F2, 2, 18) (dual of [(4194434, 2), 8388627, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(234, 133, F2, 2, 9) (dual of [(133, 2), 232, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(234, 266, F2, 9) (dual of [266, 232, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(233, 256, F2, 9) (dual of [256, 223, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(225, 256, F2, 7) (dual of [256, 231, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(8) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(234, 266, F2, 9) (dual of [266, 232, 10]-code), using
- linear OOA(2207, 4194301, F2, 2, 18) (dual of [(4194301, 2), 8388395, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2207, 8388602, F2, 18) (dual of [8388602, 8388395, 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 2-folding [i] based on linear OA(2207, 8388602, F2, 18) (dual of [8388602, 8388395, 19]-code), using
- linear OOA(234, 133, F2, 2, 9) (dual of [(133, 2), 232, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA 2-folding [i] based on linear OOA(2241, 4194434, F2, 2, 18) (dual of [(4194434, 2), 8388627, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2241, 2097217, F2, 4, 18) (dual of [(2097217, 4), 8388627, 19]-NRT-code), using
(223, 241, large)-Net in Base 2 — Upper bound on s
There is no (223, 241, large)-net in base 2, because
- 16 times m-reduction [i] would yield (223, 225, large)-net in base 2, but