Best Known (117−12, 117, s)-Nets in Base 4
(117−12, 117, 1398118)-Net over F4 — Constructive and digital
Digital (105, 117, 1398118)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 18)-net over F4, using
- digital (96, 108, 1398100)-net over F4, using
- net defined by OOA [i] based on linear OOA(4108, 1398100, F4, 12, 12) (dual of [(1398100, 12), 16777092, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(4108, 8388600, F4, 12) (dual of [8388600, 8388492, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(4108, large, F4, 12) (dual of [large, large−108, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(4108, large, F4, 12) (dual of [large, large−108, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(4108, 8388600, F4, 12) (dual of [8388600, 8388492, 13]-code), using
- net defined by OOA [i] based on linear OOA(4108, 1398100, F4, 12, 12) (dual of [(1398100, 12), 16777092, 13]-NRT-code), using
(117−12, 117, large)-Net over F4 — Digital
Digital (105, 117, large)-net over F4, using
- 43 times duplication [i] based on digital (102, 114, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4114, large, F4, 12) (dual of [large, large−114, 13]-code), using
- 6 times code embedding in larger space [i] based on linear OA(4108, large, F4, 12) (dual of [large, large−108, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 6 times code embedding in larger space [i] based on linear OA(4108, large, F4, 12) (dual of [large, large−108, 13]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4114, large, F4, 12) (dual of [large, large−114, 13]-code), using
(117−12, 117, large)-Net in Base 4 — Upper bound on s
There is no (105, 117, large)-net in base 4, because
- 10 times m-reduction [i] would yield (105, 107, large)-net in base 4, but