Best Known (103, 112, s)-Nets in Base 5
(103, 112, 5170873)-Net over F5 — Constructive and digital
Digital (103, 112, 5170873)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (26, 30, 976573)-net over F5, using
- net defined by OOA [i] based on linear OOA(530, 976573, F5, 4, 4) (dual of [(976573, 4), 3906262, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(530, 976573, F5, 3, 4) (dual of [(976573, 3), 2929689, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(530, 1953146, F5, 4) (dual of [1953146, 1953116, 5]-code), using
- 1 times code embedding in larger space [i] based on linear OA(529, 1953145, F5, 4) (dual of [1953145, 1953116, 5]-code), using
- construction X4 applied to Ce(3) ⊂ Ce(1) [i] based on
- linear OA(528, 1953125, F5, 4) (dual of [1953125, 1953097, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(510, 1953125, F5, 2) (dual of [1953125, 1953115, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(519, 20, F5, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,5)), using
- dual of repetition code with length 20 [i]
- linear OA(51, 20, F5, 1) (dual of [20, 19, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(3) ⊂ Ce(1) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(529, 1953145, F5, 4) (dual of [1953145, 1953116, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(530, 1953146, F5, 4) (dual of [1953146, 1953116, 5]-code), using
- appending kth column [i] based on linear OOA(530, 976573, F5, 3, 4) (dual of [(976573, 3), 2929689, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(530, 976573, F5, 4, 4) (dual of [(976573, 4), 3906262, 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 (26, 30, 976573)-net over F5, using
(103, 112, large)-Net over F5 — Digital
Digital (103, 112, large)-net over F5, using
- 53 times duplication [i] based on digital (100, 109, large)-net over F5, using
- t-expansion [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
- t-expansion [i] based on digital (96, 109, large)-net over F5, using
(103, 112, large)-Net in Base 5 — Upper bound on s
There is no (103, 112, large)-net in base 5, because
- 7 times m-reduction [i] would yield (103, 105, large)-net in base 5, but