Best Known (87, 109, s)-Nets in Base 16
(87, 109, 95328)-Net over F16 — Constructive and digital
Digital (87, 109, 95328)-net over F16, using
- 161 times duplication [i] based on digital (86, 108, 95328)-net over F16, using
- net defined by OOA [i] based on linear OOA(16108, 95328, F16, 22, 22) (dual of [(95328, 22), 2097108, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(16108, 1048608, F16, 22) (dual of [1048608, 1048500, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16107, 1048607, F16, 22) (dual of [1048607, 1048500, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(16101, 1048576, F16, 22) (dual of [1048576, 1048475, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(166, 31, F16, 4) (dual of [31, 25, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- 1 times truncation [i] based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(16107, 1048607, F16, 22) (dual of [1048607, 1048500, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(16108, 1048608, F16, 22) (dual of [1048608, 1048500, 23]-code), using
- net defined by OOA [i] based on linear OOA(16108, 95328, F16, 22, 22) (dual of [(95328, 22), 2097108, 23]-NRT-code), using
(87, 109, 1048611)-Net over F16 — Digital
Digital (87, 109, 1048611)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16109, 1048611, F16, 22) (dual of [1048611, 1048502, 23]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(16107, 1048607, F16, 22) (dual of [1048607, 1048500, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(16101, 1048576, F16, 22) (dual of [1048576, 1048475, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(166, 31, F16, 4) (dual of [31, 25, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- 1 times truncation [i] based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(16107, 1048609, F16, 21) (dual of [1048609, 1048502, 22]-code), using Gilbert–Varšamov bound and bm = 16107 > Vbs−1(k−1) = 353090 277226 406078 116575 286121 936373 062540 042872 065185 035191 613776 459335 310062 216331 505724 226055 090467 480985 355830 227884 218171 [i]
- linear OA(160, 2, F16, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(16107, 1048607, F16, 22) (dual of [1048607, 1048500, 23]-code), using
- construction X with Varšamov bound [i] based on
(87, 109, large)-Net in Base 16 — Upper bound on s
There is no (87, 109, large)-net in base 16, because
- 20 times m-reduction [i] would yield (87, 89, large)-net in base 16, but