Best Known (55, 55+13, s)-Nets in Base 2
(55, 55+13, 343)-Net over F2 — Constructive and digital
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
(55, 55+13, 607)-Net over F2 — Digital
Digital (55, 68, 607)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(268, 607, F2, 3, 13) (dual of [(607, 3), 1753, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(268, 686, F2, 3, 13) (dual of [(686, 3), 1990, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(268, 2058, F2, 13) (dual of [2058, 1990, 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 3-folding [i] based on linear OA(268, 2058, F2, 13) (dual of [2058, 1990, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(268, 686, F2, 3, 13) (dual of [(686, 3), 1990, 14]-NRT-code), using
(55, 55+13, 6873)-Net in Base 2 — Upper bound on s
There is no (55, 68, 6874)-net in base 2, because
- 1 times m-reduction [i] would yield (55, 67, 6874)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 147 619283 951530 730620 > 267 [i]