Best Known (123−15, 123, s)-Nets in Base 5
(123−15, 123, 1198371)-Net over F5 — Constructive and digital
Digital (108, 123, 1198371)-net over F5, using
- 53 times duplication [i] based on digital (105, 120, 1198371)-net over F5, using
- net defined by OOA [i] based on linear OOA(5120, 1198371, F5, 15, 15) (dual of [(1198371, 15), 17975445, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5120, 8388598, F5, 15) (dual of [8388598, 8388478, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5120, large, F5, 15) (dual of [large, large−120, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(5120, large, F5, 15) (dual of [large, large−120, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5120, 8388598, F5, 15) (dual of [8388598, 8388478, 16]-code), using
- net defined by OOA [i] based on linear OOA(5120, 1198371, F5, 15, 15) (dual of [(1198371, 15), 17975445, 16]-NRT-code), using
(123−15, 123, 5139373)-Net over F5 — Digital
Digital (108, 123, 5139373)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5123, 5139373, F5, 15) (dual of [5139373, 5139250, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5123, large, F5, 15) (dual of [large, large−123, 16]-code), using
- 3 times code embedding in larger space [i] based on linear OA(5120, large, F5, 15) (dual of [large, large−120, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 3 times code embedding in larger space [i] based on linear OA(5120, large, F5, 15) (dual of [large, large−120, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5123, large, F5, 15) (dual of [large, large−123, 16]-code), using
(123−15, 123, large)-Net in Base 5 — Upper bound on s
There is no (108, 123, large)-net in base 5, because
- 13 times m-reduction [i] would yield (108, 110, large)-net in base 5, but