Best Known (105−18, 105, s)-Nets in Base 5
(105−18, 105, 8683)-Net over F5 — Constructive and digital
Digital (87, 105, 8683)-net over F5, using
- 52 times duplication [i] based on digital (85, 103, 8683)-net over F5, using
- net defined by OOA [i] based on linear OOA(5103, 8683, F5, 18, 18) (dual of [(8683, 18), 156191, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(5103, 78147, F5, 18) (dual of [78147, 78044, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5103, 78150, F5, 18) (dual of [78150, 78047, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(599, 78125, F5, 18) (dual of [78125, 78026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(54, 25, F5, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,5)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(5103, 78150, F5, 18) (dual of [78150, 78047, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(5103, 78147, F5, 18) (dual of [78147, 78044, 19]-code), using
- net defined by OOA [i] based on linear OOA(5103, 8683, F5, 18, 18) (dual of [(8683, 18), 156191, 19]-NRT-code), using
(105−18, 105, 59389)-Net over F5 — Digital
Digital (87, 105, 59389)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5105, 59389, F5, 18) (dual of [59389, 59284, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5105, 78155, F5, 18) (dual of [78155, 78050, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(599, 78125, F5, 18) (dual of [78125, 78026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(571, 78125, F5, 13) (dual of [78125, 78054, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(5105, 78155, F5, 18) (dual of [78155, 78050, 19]-code), using
(105−18, 105, large)-Net in Base 5 — Upper bound on s
There is no (87, 105, large)-net in base 5, because
- 16 times m-reduction [i] would yield (87, 89, large)-net in base 5, but