Best Known (126−21, 126, s)-Nets in Base 16
(126−21, 126, 1677720)-Net over F16 — Constructive and digital
Digital (105, 126, 1677720)-net over F16, using
- 164 times duplication [i] based on digital (101, 122, 1677720)-net over F16, using
- net defined by OOA [i] based on linear OOA(16122, 1677720, F16, 22, 21) (dual of [(1677720, 22), 36909718, 22]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(16122, 8388601, F16, 2, 21) (dual of [(8388601, 2), 16777080, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(16122, 8388602, F16, 2, 21) (dual of [(8388602, 2), 16777082, 22]-NRT-code), using
- trace code [i] based on linear OOA(25661, 4194301, F256, 2, 21) (dual of [(4194301, 2), 8388541, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25661, 8388602, F256, 21) (dual of [8388602, 8388541, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- OOA 2-folding [i] based on linear OA(25661, 8388602, F256, 21) (dual of [8388602, 8388541, 22]-code), using
- trace code [i] based on linear OOA(25661, 4194301, F256, 2, 21) (dual of [(4194301, 2), 8388541, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(16122, 8388602, F16, 2, 21) (dual of [(8388602, 2), 16777082, 22]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(16122, 8388601, F16, 2, 21) (dual of [(8388601, 2), 16777080, 22]-NRT-code), using
- net defined by OOA [i] based on linear OOA(16122, 1677720, F16, 22, 21) (dual of [(1677720, 22), 36909718, 22]-NRT-code), using
(126−21, 126, large)-Net over F16 — Digital
Digital (105, 126, large)-net over F16, using
- t-expansion [i] based on digital (104, 126, large)-net over F16, using
- 1 times m-reduction [i] based on digital (104, 127, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16127, large, F16, 23) (dual of [large, large−127, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16127, large, F16, 23) (dual of [large, large−127, 24]-code), using
- 1 times m-reduction [i] based on digital (104, 127, large)-net over F16, using
(126−21, 126, large)-Net in Base 16 — Upper bound on s
There is no (105, 126, large)-net in base 16, because
- 19 times m-reduction [i] would yield (105, 107, large)-net in base 16, but