Best Known (104−21, 104, s)-Nets in Base 16
(104−21, 104, 104860)-Net over F16 — Constructive and digital
Digital (83, 104, 104860)-net over F16, using
- 163 times duplication [i] based on digital (80, 101, 104860)-net over F16, using
- net defined by OOA [i] based on linear OOA(16101, 104860, F16, 21, 21) (dual of [(104860, 21), 2201959, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(16101, 1048601, F16, 21) (dual of [1048601, 1048500, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16100, 1048600, F16, 21) (dual of [1048600, 1048500, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(1696, 1048576, F16, 21) (dual of [1048576, 1048480, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(164, 24, F16, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,16)), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(16100, 1048600, F16, 21) (dual of [1048600, 1048500, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(16101, 1048601, F16, 21) (dual of [1048601, 1048500, 22]-code), using
- net defined by OOA [i] based on linear OOA(16101, 104860, F16, 21, 21) (dual of [(104860, 21), 2201959, 22]-NRT-code), using
(104−21, 104, 1048610)-Net over F16 — Digital
Digital (83, 104, 1048610)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16104, 1048610, F16, 21) (dual of [1048610, 1048506, 22]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(16103, 1048608, F16, 21) (dual of [1048608, 1048505, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(1696, 1048576, F16, 21) (dual of [1048576, 1048480, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1671, 1048576, F16, 15) (dual of [1048576, 1048505, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(167, 32, F16, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(16103, 1048609, F16, 20) (dual of [1048609, 1048506, 21]-code), using Gilbert–Varšamov bound and bm = 16103 > Vbs−1(k−1) = 448971 910223 963428 047539 735841 154341 313090 258830 540225 268462 361422 330366 219926 157901 604689 163468 908230 576014 554056 093171 [i]
- linear OA(160, 1, F16, 0) (dual of [1, 1, 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(16103, 1048608, F16, 21) (dual of [1048608, 1048505, 22]-code), using
- construction X with Varšamov bound [i] based on
(104−21, 104, large)-Net in Base 16 — Upper bound on s
There is no (83, 104, large)-net in base 16, because
- 19 times m-reduction [i] would yield (83, 85, large)-net in base 16, but