Best Known (101−14, 101, s)-Nets in Base 2
(101−14, 101, 2343)-Net over F2 — Constructive and digital
Digital (87, 101, 2343)-net over F2, using
- net defined by OOA [i] based on linear OOA(2101, 2343, F2, 14, 14) (dual of [(2343, 14), 32701, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2101, 16401, F2, 14) (dual of [16401, 16300, 15]-code), using
- 2 times code embedding in larger space [i] based on linear OA(299, 16399, F2, 14) (dual of [16399, 16300, 15]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2100, 16400, F2, 15) (dual of [16400, 16300, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(299, 16399, F2, 14) (dual of [16399, 16300, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(2101, 16401, F2, 14) (dual of [16401, 16300, 15]-code), using
(101−14, 101, 4100)-Net over F2 — Digital
Digital (87, 101, 4100)-net over F2, using
- 21 times duplication [i] based on digital (86, 100, 4100)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2100, 4100, F2, 4, 14) (dual of [(4100, 4), 16300, 15]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2100, 16400, F2, 14) (dual of [16400, 16300, 15]-code), using
- strength reduction [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
- strength reduction [i] based on linear OA(2100, 16400, F2, 15) (dual of [16400, 16300, 16]-code), using
- OOA 4-folding [i] based on linear OA(2100, 16400, F2, 14) (dual of [16400, 16300, 15]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2100, 4100, F2, 4, 14) (dual of [(4100, 4), 16300, 15]-NRT-code), using
(101−14, 101, 74523)-Net in Base 2 — Upper bound on s
There is no (87, 101, 74524)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2 535423 099737 784686 884413 402514 > 2101 [i]