Best Known (116, 116+13, s)-Nets in Base 5
(116, 116+13, 2796210)-Net over F5 — Constructive and digital
Digital (116, 129, 2796210)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (109, 122, 2796200)-net over F5, using
- net defined by OOA [i] based on linear OOA(5122, 2796200, F5, 14, 13) (dual of [(2796200, 14), 39146678, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(5122, 8388601, F5, 2, 13) (dual of [(8388601, 2), 16777080, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(5122, 8388602, F5, 2, 13) (dual of [(8388602, 2), 16777082, 14]-NRT-code), using
- trace code [i] based on linear OOA(2561, 4194301, F25, 2, 13) (dual of [(4194301, 2), 8388541, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2561, 8388602, F25, 13) (dual of [8388602, 8388541, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- OOA 2-folding [i] based on linear OA(2561, 8388602, F25, 13) (dual of [8388602, 8388541, 14]-code), using
- trace code [i] based on linear OOA(2561, 4194301, F25, 2, 13) (dual of [(4194301, 2), 8388541, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(5122, 8388602, F5, 2, 13) (dual of [(8388602, 2), 16777082, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(5122, 8388601, F5, 2, 13) (dual of [(8388601, 2), 16777080, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(5122, 2796200, F5, 14, 13) (dual of [(2796200, 14), 39146678, 14]-NRT-code), using
- digital (1, 7, 10)-net over F5, using
(116, 116+13, large)-Net over F5 — Digital
Digital (116, 129, large)-net over F5, using
- 52 times duplication [i] based on digital (114, 127, large)-net over F5, using
- t-expansion [i] based on digital (112, 127, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5127, large, F5, 15) (dual of [large, large−127, 16]-code), using
- 7 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]
- 7 times code embedding in larger space [i] based on linear OA(5120, large, F5, 15) (dual of [large, large−120, 16]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5127, large, F5, 15) (dual of [large, large−127, 16]-code), using
- t-expansion [i] based on digital (112, 127, large)-net over F5, using
(116, 116+13, large)-Net in Base 5 — Upper bound on s
There is no (116, 129, large)-net in base 5, because
- 11 times m-reduction [i] would yield (116, 118, large)-net in base 5, but