Best Known (67, 67+20, s)-Nets in Base 16
(67, 67+20, 13110)-Net over F16 — Constructive and digital
Digital (67, 87, 13110)-net over F16, using
- 161 times duplication [i] based on digital (66, 86, 13110)-net over F16, using
- net defined by OOA [i] based on linear OOA(1686, 13110, F16, 20, 20) (dual of [(13110, 20), 262114, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(1686, 131100, F16, 20) (dual of [131100, 131014, 21]-code), using
- trace code [i] based on linear OA(25643, 65550, F256, 20) (dual of [65550, 65507, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(25629, 65536, F256, 15) (dual of [65536, 65507, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2564, 14, F256, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- Reed–Solomon code RS(252,256) [i]
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- trace code [i] based on linear OA(25643, 65550, F256, 20) (dual of [65550, 65507, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(1686, 131100, F16, 20) (dual of [131100, 131014, 21]-code), using
- net defined by OOA [i] based on linear OOA(1686, 13110, F16, 20, 20) (dual of [(13110, 20), 262114, 21]-NRT-code), using
(67, 67+20, 26214)-Net in Base 16 — Constructive
(67, 87, 26214)-net in base 16, using
- base change [i] based on digital (38, 58, 26214)-net over F64, using
- net defined by OOA [i] based on linear OOA(6458, 26214, F64, 20, 20) (dual of [(26214, 20), 524222, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(6458, 262140, F64, 20) (dual of [262140, 262082, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(6458, 262140, F64, 20) (dual of [262140, 262082, 21]-code), using
- net defined by OOA [i] based on linear OOA(6458, 26214, F64, 20, 20) (dual of [(26214, 20), 524222, 21]-NRT-code), using
(67, 67+20, 172484)-Net over F16 — Digital
Digital (67, 87, 172484)-net over F16, using
(67, 67+20, large)-Net in Base 16 — Upper bound on s
There is no (67, 87, large)-net in base 16, because
- 18 times m-reduction [i] would yield (67, 69, large)-net in base 16, but