Best Known (78, 86, s)-Nets in Base 5
(78, 86, 4194928)-Net over F5 — Constructive and digital
Digital (78, 86, 4194928)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (10, 14, 628)-net over F5, using
- net defined by OOA [i] based on linear OOA(514, 628, F5, 4, 4) (dual of [(628, 4), 2498, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(514, 628, F5, 3, 4) (dual of [(628, 3), 1870, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(514, 1256, F5, 4) (dual of [1256, 1242, 5]-code), using
- trace code [i] based on linear OA(257, 628, F25, 4) (dual of [628, 621, 5]-code), using
- construction XX applied to C1 = C([623,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([623,2]) [i] based on
- linear OA(255, 624, F25, 3) (dual of [624, 619, 4]-code or 624-cap in PG(4,25)), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(255, 624, F25, 3) (dual of [624, 619, 4]-code or 624-cap in PG(4,25)), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(257, 624, F25, 4) (dual of [624, 617, 5]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(253, 624, F25, 2) (dual of [624, 621, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([623,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([623,2]) [i] based on
- trace code [i] based on linear OA(257, 628, F25, 4) (dual of [628, 621, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(514, 1256, F5, 4) (dual of [1256, 1242, 5]-code), using
- appending kth column [i] based on linear OOA(514, 628, F5, 3, 4) (dual of [(628, 3), 1870, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(514, 628, F5, 4, 4) (dual of [(628, 4), 2498, 5]-NRT-code), using
- digital (64, 72, 4194300)-net over F5, using
- net defined by OOA [i] based on linear OOA(572, 4194300, F5, 10, 8) (dual of [(4194300, 10), 41942928, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(572, 8388601, F5, 2, 8) (dual of [(8388601, 2), 16777130, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(572, 8388602, F5, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- trace code [i] based on linear OOA(2536, 4194301, F25, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2536, 8388602, F25, 8) (dual of [8388602, 8388566, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- OOA 2-folding [i] based on linear OA(2536, 8388602, F25, 8) (dual of [8388602, 8388566, 9]-code), using
- trace code [i] based on linear OOA(2536, 4194301, F25, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(572, 8388602, F5, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(572, 8388601, F5, 2, 8) (dual of [(8388601, 2), 16777130, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(572, 4194300, F5, 10, 8) (dual of [(4194300, 10), 41942928, 9]-NRT-code), using
- digital (10, 14, 628)-net over F5, using
(78, 86, large)-Net over F5 — Digital
Digital (78, 86, large)-net over F5, using
- 55 times duplication [i] based on digital (73, 81, large)-net over F5, using
- t-expansion [i] based on digital (71, 81, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(581, large, F5, 10) (dual of [large, large−81, 11]-code), using
- strength reduction [i] based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 520−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- strength reduction [i] based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(581, large, F5, 10) (dual of [large, large−81, 11]-code), using
- t-expansion [i] based on digital (71, 81, large)-net over F5, using
(78, 86, large)-Net in Base 5 — Upper bound on s
There is no (78, 86, large)-net in base 5, because
- 6 times m-reduction [i] would yield (78, 80, large)-net in base 5, but