Best Known (108−12, 108, s)-Nets in Base 4
(108−12, 108, 1398100)-Net over F4 — Constructive and digital
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
(108−12, 108, 4194301)-Net over F4 — Digital
Digital (96, 108, 4194301)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4108, 4194301, F4, 2, 12) (dual of [(4194301, 2), 8388494, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4108, 8388602, F4, 12) (dual of [8388602, 8388494, 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
- OOA 2-folding [i] based on linear OA(4108, 8388602, F4, 12) (dual of [8388602, 8388494, 13]-code), using
(108−12, 108, large)-Net in Base 4 — Upper bound on s
There is no (96, 108, large)-net in base 4, because
- 10 times m-reduction [i] would yield (96, 98, large)-net in base 4, but