Best Known (66−17, 66, s)-Nets in Base 5
(66−17, 66, 391)-Net over F5 — Constructive and digital
Digital (49, 66, 391)-net over F5, using
- net defined by OOA [i] based on linear OOA(566, 391, F5, 17, 17) (dual of [(391, 17), 6581, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(566, 3129, F5, 17) (dual of [3129, 3063, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(566, 3130, F5, 17) (dual of [3130, 3064, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(566, 3130, F5, 17) (dual of [3130, 3064, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(566, 3129, F5, 17) (dual of [3129, 3063, 18]-code), using
(66−17, 66, 1706)-Net over F5 — Digital
Digital (49, 66, 1706)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(566, 1706, F5, 17) (dual of [1706, 1640, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using
(66−17, 66, 449527)-Net in Base 5 — Upper bound on s
There is no (49, 66, 449528)-net in base 5, because
- 1 times m-reduction [i] would yield (49, 65, 449528)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 2710 540495 897909 978639 592727 698894 083760 538625 > 565 [i]