Best Known (104−15, 104, s)-Nets in Base 2
(104−15, 104, 2343)-Net over F2 — Constructive and digital
Digital (89, 104, 2343)-net over F2, using
- 22 times duplication [i] based on digital (87, 102, 2343)-net over F2, using
- net defined by OOA [i] based on linear OOA(2102, 2343, F2, 15, 15) (dual of [(2343, 15), 35043, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2102, 16402, F2, 15) (dual of [16402, 16300, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2100, 16400, F2, 15) (dual of [16400, 16300, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(299, 16384, F2, 15) (dual of [16384, 16285, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(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(14) ⊂ Ce(12) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2100, 16400, F2, 15) (dual of [16400, 16300, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2102, 16402, F2, 15) (dual of [16402, 16300, 16]-code), using
- net defined by OOA [i] based on linear OOA(2102, 2343, F2, 15, 15) (dual of [(2343, 15), 35043, 16]-NRT-code), using
(104−15, 104, 4101)-Net over F2 — Digital
Digital (89, 104, 4101)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2104, 4101, F2, 4, 15) (dual of [(4101, 4), 16300, 16]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2100, 4100, F2, 4, 15) (dual of [(4100, 4), 16300, 16]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2100, 16400, F2, 15) (dual of [16400, 16300, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(299, 16384, F2, 15) (dual of [16384, 16285, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(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(14) ⊂ Ce(12) [i] based on
- OOA 4-folding [i] based on linear OA(2100, 16400, F2, 15) (dual of [16400, 16300, 16]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2100, 4100, F2, 4, 15) (dual of [(4100, 4), 16300, 16]-NRT-code), using
(104−15, 104, 90847)-Net in Base 2 — Upper bound on s
There is no (89, 104, 90848)-net in base 2, because
- 1 times m-reduction [i] would yield (89, 103, 90848)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 10 141720 943924 138184 203487 830793 > 2103 [i]