Best Known (107−9, 107, s)-Nets in Base 5
(107−9, 107, 4389616)-Net over F5 — Constructive and digital
Digital (98, 107, 4389616)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (21, 25, 195316)-net over F5, using
- net defined by OOA [i] based on linear OOA(525, 195316, F5, 4, 4) (dual of [(195316, 4), 781239, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(525, 195316, F5, 3, 4) (dual of [(195316, 3), 585923, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(525, 390632, F5, 4) (dual of [390632, 390607, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(525, 390633, F5, 4) (dual of [390633, 390608, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(525, 390625, F5, 4) (dual of [390625, 390600, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(517, 390625, F5, 3) (dual of [390625, 390608, 4]-code or 390625-cap in PG(16,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(525, 390633, F5, 4) (dual of [390633, 390608, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(525, 390632, F5, 4) (dual of [390632, 390607, 5]-code), using
- appending kth column [i] based on linear OOA(525, 195316, F5, 3, 4) (dual of [(195316, 3), 585923, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(525, 195316, F5, 4, 4) (dual of [(195316, 4), 781239, 5]-NRT-code), using
- digital (73, 82, 4194300)-net over F5, using
- net defined by OOA [i] based on linear OOA(582, 4194300, F5, 10, 9) (dual of [(4194300, 10), 41942918, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(582, 8388601, F5, 2, 9) (dual of [(8388601, 2), 16777120, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(582, 8388602, F5, 2, 9) (dual of [(8388602, 2), 16777122, 10]-NRT-code), using
- trace code [i] based on linear OOA(2541, 4194301, F25, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2541, 8388602, F25, 9) (dual of [8388602, 8388561, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2541, large, F25, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2541, large, F25, 9) (dual of [large, large−41, 10]-code), using
- OOA 2-folding [i] based on linear OA(2541, 8388602, F25, 9) (dual of [8388602, 8388561, 10]-code), using
- trace code [i] based on linear OOA(2541, 4194301, F25, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(582, 8388602, F5, 2, 9) (dual of [(8388602, 2), 16777122, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(582, 8388601, F5, 2, 9) (dual of [(8388601, 2), 16777120, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(582, 4194300, F5, 10, 9) (dual of [(4194300, 10), 41942918, 10]-NRT-code), using
- digital (21, 25, 195316)-net over F5, using
(107−9, 107, large)-Net over F5 — Digital
Digital (98, 107, large)-net over F5, using
- t-expansion [i] based on digital (96, 107, large)-net over F5, using
- 2 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
- 2 times m-reduction [i] based on digital (96, 109, large)-net over F5, using
(107−9, 107, large)-Net in Base 5 — Upper bound on s
There is no (98, 107, large)-net in base 5, because
- 7 times m-reduction [i] would yield (98, 100, large)-net in base 5, but