Best Known (69, 69+18, s)-Nets in Base 5
(69, 69+18, 1737)-Net over F5 — Constructive and digital
Digital (69, 87, 1737)-net over F5, using
- 51 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(586, 1737, F5, 18, 18) (dual of [(1737, 18), 31180, 19]-NRT-code), using
(69, 69+18, 9704)-Net over F5 — Digital
Digital (69, 87, 9704)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(587, 9704, F5, 18) (dual of [9704, 9617, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(587, 15639, F5, 18) (dual of [15639, 15552, 19]-code), using
- 1 times code embedding in larger space [i] 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
- 1 times code embedding in larger space [i] based on linear OA(586, 15638, F5, 18) (dual of [15638, 15552, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(587, 15639, F5, 18) (dual of [15639, 15552, 19]-code), using
(69, 69+18, 5921081)-Net in Base 5 — Upper bound on s
There is no (69, 87, 5921082)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 6 462353 266302 229369 034282 635155 420539 817454 760583 073836 418025 > 587 [i]