Best Known (108−12, 108, s)-Nets in Base 5
(108−12, 108, 1398308)-Net over F5 — Constructive and digital
Digital (96, 108, 1398308)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (11, 17, 208)-net over F5, using
- net defined by OOA [i] based on linear OOA(517, 208, F5, 6, 6) (dual of [(208, 6), 1231, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(517, 624, F5, 6) (dual of [624, 607, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(517, 624, F5, 6) (dual of [624, 607, 7]-code), using
- net defined by OOA [i] based on linear OOA(517, 208, F5, 6, 6) (dual of [(208, 6), 1231, 7]-NRT-code), using
- digital (79, 91, 1398100)-net over F5, using
- net defined by OOA [i] based on linear OOA(591, 1398100, F5, 12, 12) (dual of [(1398100, 12), 16777109, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(591, 8388600, F5, 12) (dual of [8388600, 8388509, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(591, large, F5, 12) (dual of [large, large−91, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(591, large, F5, 12) (dual of [large, large−91, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(591, 8388600, F5, 12) (dual of [8388600, 8388509, 13]-code), using
- net defined by OOA [i] based on linear OOA(591, 1398100, F5, 12, 12) (dual of [(1398100, 12), 16777109, 13]-NRT-code), using
- digital (11, 17, 208)-net over F5, using
(108−12, 108, large)-Net over F5 — Digital
Digital (96, 108, large)-net over F5, using
- 1 times m-reduction [i] based on digital (96, 109, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5109, large, F5, 13) (dual of [large, large−109, 14]-code), using
- 8 times code embedding in larger space [i] based on linear OA(5101, large, F5, 13) (dual of [large, large−101, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 520−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 8 times code embedding in larger space [i] based on linear OA(5101, large, F5, 13) (dual of [large, large−101, 14]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5109, large, F5, 13) (dual of [large, large−109, 14]-code), using
(108−12, 108, large)-Net in Base 5 — Upper bound on s
There is no (96, 108, large)-net in base 5, because
- 10 times m-reduction [i] would yield (96, 98, large)-net in base 5, but