Best Known (71, 89, s)-Nets in Base 5
(71, 89, 1738)-Net over F5 — Constructive and digital
Digital (71, 89, 1738)-net over F5, using
- net defined by OOA [i] based on linear OOA(589, 1738, F5, 18, 18) (dual of [(1738, 18), 31195, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(589, 15642, F5, 18) (dual of [15642, 15553, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(589, 15647, F5, 18) (dual of [15647, 15558, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(585, 15625, F5, 18) (dual of [15625, 15540, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(54, 22, F5, 3) (dual of [22, 18, 4]-code or 22-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(589, 15647, F5, 18) (dual of [15647, 15558, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(589, 15642, F5, 18) (dual of [15642, 15553, 19]-code), using
(71, 89, 11869)-Net over F5 — Digital
Digital (71, 89, 11869)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(589, 11869, F5, 18) (dual of [11869, 11780, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(589, 15647, F5, 18) (dual of [15647, 15558, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(585, 15625, F5, 18) (dual of [15625, 15540, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(54, 22, F5, 3) (dual of [22, 18, 4]-code or 22-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(589, 15647, F5, 18) (dual of [15647, 15558, 19]-code), using
(71, 89, large)-Net in Base 5 — Upper bound on s
There is no (71, 89, large)-net in base 5, because
- 16 times m-reduction [i] would yield (71, 73, large)-net in base 5, but