Best Known (65−16, 65, s)-Nets in Base 5
(65−16, 65, 393)-Net over F5 — Constructive and digital
Digital (49, 65, 393)-net over F5, using
- net defined by OOA [i] based on linear OOA(565, 393, F5, 16, 16) (dual of [(393, 16), 6223, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(565, 3144, F5, 16) (dual of [3144, 3079, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- 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(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,5)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- OA 8-folding and stacking [i] based on linear OA(565, 3144, F5, 16) (dual of [3144, 3079, 17]-code), using
(65−16, 65, 2360)-Net over F5 — Digital
Digital (49, 65, 2360)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(565, 2360, F5, 16) (dual of [2360, 2295, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(565, 3144, F5, 16) (dual of [3144, 3079, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- 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(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,5)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(565, 3144, F5, 16) (dual of [3144, 3079, 17]-code), using
(65−16, 65, 449527)-Net in Base 5 — Upper bound on s
There is no (49, 65, 449528)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 2710 540495 897909 978639 592727 698894 083760 538625 > 565 [i]