Best Known (68, 68+18, s)-Nets in Base 5
(68, 68+18, 1737)-Net over F5 — Constructive and digital
Digital (68, 86, 1737)-net over F5, using
- net defined by OOA [i] based on linear OOA(586, 1737, F5, 18, 18) (dual of [(1737, 18), 31180, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(586, 15633, F5, 18) (dual of [15633, 15547, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(586, 15638, F5, 18) (dual of [15638, 15552, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [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(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(51, 13, F5, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(586, 15638, F5, 18) (dual of [15638, 15552, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(586, 15633, F5, 18) (dual of [15633, 15547, 19]-code), using
(68, 68+18, 8774)-Net over F5 — Digital
Digital (68, 86, 8774)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(586, 8774, F5, 18) (dual of [8774, 8688, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(586, 15638, F5, 18) (dual of [15638, 15552, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [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(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(51, 13, F5, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(586, 15638, F5, 18) (dual of [15638, 15552, 19]-code), using
(68, 68+18, 4951509)-Net in Base 5 — Upper bound on s
There is no (68, 86, 4951510)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1 292470 880168 717568 357663 173865 665799 067424 588036 536690 210649 > 586 [i]