Best Known (71−13, 71, s)-Nets in Base 2
(71−13, 71, 343)-Net over F2 — Constructive and digital
Digital (58, 71, 343)-net over F2, using
- 23 times duplication [i] based on digital (55, 68, 343)-net over F2, using
- net defined by OOA [i] based on linear OOA(268, 343, F2, 13, 13) (dual of [(343, 13), 4391, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(268, 2059, F2, 13) (dual of [2059, 1991, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(268, 2060, F2, 13) (dual of [2060, 1992, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(267, 2048, F2, 13) (dual of [2048, 1981, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(21, 12, F2, 1) (dual of [12, 11, 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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(268, 2060, F2, 13) (dual of [2060, 1992, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(268, 2059, F2, 13) (dual of [2059, 1991, 14]-code), using
- net defined by OOA [i] based on linear OOA(268, 343, F2, 13, 13) (dual of [(343, 13), 4391, 14]-NRT-code), using
(71−13, 71, 688)-Net over F2 — Digital
Digital (58, 71, 688)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(271, 688, F2, 3, 13) (dual of [(688, 3), 1993, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(271, 2064, F2, 13) (dual of [2064, 1993, 14]-code), using
- 2 times code embedding in larger space [i] based on linear OA(269, 2062, F2, 13) (dual of [2062, 1993, 14]-code), using
- adding a parity check bit [i] based on linear OA(268, 2061, F2, 12) (dual of [2061, 1993, 13]-code), using
- construction X4 applied to C([0,12]) ⊂ C([1,10]) [i] based on
- linear OA(267, 2047, F2, 13) (dual of [2047, 1980, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(255, 2047, F2, 10) (dual of [2047, 1992, 11]-code), using 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(213, 14, F2, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,2)), using
- dual of repetition code with length 14 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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 C([0,12]) ⊂ C([1,10]) [i] based on
- adding a parity check bit [i] based on linear OA(268, 2061, F2, 12) (dual of [2061, 1993, 13]-code), using
- 2 times code embedding in larger space [i] based on linear OA(269, 2062, F2, 13) (dual of [2062, 1993, 14]-code), using
- OOA 3-folding [i] based on linear OA(271, 2064, F2, 13) (dual of [2064, 1993, 14]-code), using
(71−13, 71, 9724)-Net in Base 2 — Upper bound on s
There is no (58, 71, 9725)-net in base 2, because
- 1 times m-reduction [i] would yield (58, 70, 9725)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1181 084140 722516 319496 > 270 [i]