Best Known (66, 66+10, s)-Nets in Base 16
(66, 66+10, 3551282)-Net over F16 — Constructive and digital
Digital (66, 76, 3551282)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (15, 20, 195842)-net over F16, using
- net defined by OOA [i] based on linear OOA(1620, 195842, F16, 6, 5) (dual of [(195842, 6), 1175032, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(1620, 195843, F16, 2, 5) (dual of [(195843, 2), 391666, 6]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(161, 65281, F16, 2, 1) (dual of [(65281, 2), 130561, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(161, s, F16, 2, 1) with arbitrarily large s, using
- appending 1 arbitrary column [i] based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- discarding factors / shortening the dual code based on linear OOA(161, s, F16, 2, 1) with arbitrarily large s, using
- linear OOA(165, 65281, F16, 2, 2) (dual of [(65281, 2), 130557, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(165, 69905, F16, 2, 2) (dual of [(69905, 2), 139805, 3]-NRT-code), using
- appending kth column [i] based on linear OA(165, 69905, F16, 2) (dual of [69905, 69900, 3]-code), using
- Hamming code H(5,16) [i]
- appending kth column [i] based on linear OA(165, 69905, F16, 2) (dual of [69905, 69900, 3]-code), using
- discarding factors / shortening the dual code based on linear OOA(165, 69905, F16, 2, 2) (dual of [(69905, 2), 139805, 3]-NRT-code), using
- linear OOA(1614, 65281, F16, 2, 5) (dual of [(65281, 2), 130548, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(1614, 130562, F16, 5) (dual of [130562, 130548, 6]-code), using
- trace code [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- OOA 2-folding [i] based on linear OA(1614, 130562, F16, 5) (dual of [130562, 130548, 6]-code), using
- linear OOA(161, 65281, F16, 2, 1) (dual of [(65281, 2), 130561, 2]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(1620, 195843, F16, 2, 5) (dual of [(195843, 2), 391666, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1620, 195842, F16, 6, 5) (dual of [(195842, 6), 1175032, 6]-NRT-code), using
- digital (46, 56, 3355440)-net over F16, using
- trace code for nets [i] based on digital (18, 28, 1677720)-net over F256, using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
- trace code for nets [i] based on digital (18, 28, 1677720)-net over F256, using
- digital (15, 20, 195842)-net over F16, using
(66, 66+10, large)-Net over F16 — Digital
Digital (66, 76, large)-net over F16, using
- t-expansion [i] based on digital (65, 76, large)-net over F16, using
- 3 times m-reduction [i] based on digital (65, 79, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1679, large, F16, 14) (dual of [large, large−79, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1679, large, F16, 14) (dual of [large, large−79, 15]-code), using
- 3 times m-reduction [i] based on digital (65, 79, large)-net over F16, using
(66, 66+10, large)-Net in Base 16 — Upper bound on s
There is no (66, 76, large)-net in base 16, because
- 8 times m-reduction [i] would yield (66, 68, large)-net in base 16, but