Best Known (58−11, 58, s)-Nets in Base 2
(58−11, 58, 412)-Net over F2 — Constructive and digital
Digital (47, 58, 412)-net over F2, using
- 21 times duplication [i] based on digital (46, 57, 412)-net over F2, using
- net defined by OOA [i] based on linear OOA(257, 412, F2, 11, 11) (dual of [(412, 11), 4475, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(257, 2061, F2, 11) (dual of [2061, 2004, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(256, 2048, F2, 11) (dual of [2048, 1992, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(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(10) ⊂ Ce(8) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(257, 2061, F2, 11) (dual of [2061, 2004, 12]-code), using
- net defined by OOA [i] based on linear OOA(257, 412, F2, 11, 11) (dual of [(412, 11), 4475, 12]-NRT-code), using
(58−11, 58, 687)-Net over F2 — Digital
Digital (47, 58, 687)-net over F2, using
- 21 times duplication [i] based on digital (46, 57, 687)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(257, 687, F2, 3, 11) (dual of [(687, 3), 2004, 12]-NRT-code), using
- OOA 3-folding [i] based on linear OA(257, 2061, F2, 11) (dual of [2061, 2004, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(256, 2048, F2, 11) (dual of [2048, 1992, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(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(10) ⊂ Ce(8) [i] based on
- OOA 3-folding [i] based on linear OA(257, 2061, F2, 11) (dual of [2061, 2004, 12]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(257, 687, F2, 3, 11) (dual of [(687, 3), 2004, 12]-NRT-code), using
(58−11, 58, 7033)-Net in Base 2 — Upper bound on s
There is no (47, 58, 7034)-net in base 2, because
- 1 times m-reduction [i] would yield (47, 57, 7034)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 144208 067196 986838 > 257 [i]