Best Known (70−13, 70, s)-Nets in Base 2
(70−13, 70, 343)-Net over F2 — Constructive and digital
Digital (57, 70, 343)-net over F2, using
- 22 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
(70−13, 70, 687)-Net over F2 — Digital
Digital (57, 70, 687)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(270, 687, F2, 3, 13) (dual of [(687, 3), 1991, 14]-NRT-code), using
- 21 times duplication [i] based on linear OOA(269, 687, F2, 3, 13) (dual of [(687, 3), 1992, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(269, 2061, F2, 13) (dual of [2061, 1992, 14]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(269, 2062, F2, 13) (dual of [2062, 1993, 14]-code), using
- OOA 3-folding [i] based on linear OA(269, 2061, F2, 13) (dual of [2061, 1992, 14]-code), using
- 21 times duplication [i] based on linear OOA(269, 687, F2, 3, 13) (dual of [(687, 3), 1992, 14]-NRT-code), using
(70−13, 70, 8662)-Net in Base 2 — Upper bound on s
There is no (57, 70, 8663)-net in base 2, because
- 1 times m-reduction [i] would yield (57, 69, 8663)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 590 516156 632990 606481 > 269 [i]