Best Known (38, 47, s)-Nets in Base 2
(38, 47, 515)-Net over F2 — Constructive and digital
Digital (38, 47, 515)-net over F2, using
- 21 times duplication [i] based on digital (37, 46, 515)-net over F2, using
- net defined by OOA [i] based on linear OOA(246, 515, F2, 9, 9) (dual of [(515, 9), 4589, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(246, 515, F2, 8, 9) (dual of [(515, 8), 4074, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [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
- OOA 4-folding and stacking with additional row [i] based on linear OA(246, 2061, F2, 9) (dual of [2061, 2015, 10]-code), using
- appending kth column [i] based on linear OOA(246, 515, F2, 8, 9) (dual of [(515, 8), 4074, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(246, 515, F2, 9, 9) (dual of [(515, 9), 4589, 10]-NRT-code), using
(38, 47, 687)-Net over F2 — Digital
Digital (38, 47, 687)-net over F2, using
- 21 times duplication [i] based on digital (37, 46, 687)-net over F2, using
- net defined by OOA [i] based on linear OOA(246, 687, F2, 9, 9) (dual of [(687, 9), 6137, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(246, 687, F2, 8, 9) (dual of [(687, 8), 5450, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(246, 687, F2, 3, 9) (dual of [(687, 3), 2015, 10]-NRT-code), using
- OOA 3-folding [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
- OOA 3-folding [i] based on linear OA(246, 2061, F2, 9) (dual of [2061, 2015, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(246, 687, F2, 3, 9) (dual of [(687, 3), 2015, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(246, 687, F2, 8, 9) (dual of [(687, 8), 5450, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(246, 687, F2, 9, 9) (dual of [(687, 9), 6137, 10]-NRT-code), using
(38, 47, 6405)-Net in Base 2 — Upper bound on s
There is no (38, 47, 6406)-net in base 2, because
- 1 times m-reduction [i] would yield (38, 46, 6406)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 70 408778 725301 > 246 [i]