Best Known (73, 73+12, s)-Nets in Base 2
(73, 73+12, 2733)-Net over F2 — Constructive and digital
Digital (73, 85, 2733)-net over F2, using
- net defined by OOA [i] based on linear OOA(285, 2733, F2, 12, 12) (dual of [(2733, 12), 32711, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(285, 16398, F2, 12) (dual of [16398, 16313, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(285, 16399, F2, 12) (dual of [16399, 16314, 13]-code), using
- 1 times truncation [i] based on linear OA(286, 16400, F2, 13) (dual of [16400, 16314, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(285, 16384, F2, 13) (dual of [16384, 16299, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(271, 16384, F2, 11) (dual of [16384, 16313, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(215, 16, F2, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,2)), using
- dual of repetition code with length 16 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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(12) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(286, 16400, F2, 13) (dual of [16400, 16314, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(285, 16399, F2, 12) (dual of [16399, 16314, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(285, 16398, F2, 12) (dual of [16398, 16313, 13]-code), using
(73, 73+12, 4573)-Net over F2 — Digital
Digital (73, 85, 4573)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(285, 4573, F2, 3, 12) (dual of [(4573, 3), 13634, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(285, 5466, F2, 3, 12) (dual of [(5466, 3), 16313, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(285, 16398, F2, 12) (dual of [16398, 16313, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(285, 16399, F2, 12) (dual of [16399, 16314, 13]-code), using
- 1 times truncation [i] based on linear OA(286, 16400, F2, 13) (dual of [16400, 16314, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(285, 16384, F2, 13) (dual of [16384, 16299, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(271, 16384, F2, 11) (dual of [16384, 16313, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(215, 16, F2, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,2)), using
- dual of repetition code with length 16 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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(12) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(286, 16400, F2, 13) (dual of [16400, 16314, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(285, 16399, F2, 12) (dual of [16399, 16314, 13]-code), using
- OOA 3-folding [i] based on linear OA(285, 16398, F2, 12) (dual of [16398, 16313, 13]-code), using
- discarding factors / shortening the dual code based on linear OOA(285, 5466, F2, 3, 12) (dual of [(5466, 3), 16313, 13]-NRT-code), using
(73, 73+12, 55048)-Net in Base 2 — Upper bound on s
There is no (73, 85, 55049)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 38 687120 065242 189499 798820 > 285 [i]