Best Known (86, 109, s)-Nets in Base 5
(86, 109, 1420)-Net over F5 — Constructive and digital
Digital (86, 109, 1420)-net over F5, using
- net defined by OOA [i] based on linear OOA(5109, 1420, F5, 23, 23) (dual of [(1420, 23), 32551, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5109, 15621, F5, 23) (dual of [15621, 15512, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5109, 15625, F5, 23) (dual of [15625, 15516, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(5109, 15625, F5, 23) (dual of [15625, 15516, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5109, 15621, F5, 23) (dual of [15621, 15512, 24]-code), using
(86, 109, 8519)-Net over F5 — Digital
Digital (86, 109, 8519)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5109, 8519, F5, 23) (dual of [8519, 8410, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5109, 15625, F5, 23) (dual of [15625, 15516, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(5109, 15625, F5, 23) (dual of [15625, 15516, 24]-code), using
(86, 109, large)-Net in Base 5 — Upper bound on s
There is no (86, 109, large)-net in base 5, because
- 21 times m-reduction [i] would yield (86, 88, large)-net in base 5, but