Best Known (25−9, 25, s)-Nets in Base 64
(25−9, 25, 65536)-Net over F64 — Constructive and digital
Digital (16, 25, 65536)-net over F64, using
- net defined by OOA [i] based on linear OOA(6425, 65536, F64, 9, 9) (dual of [(65536, 9), 589799, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6425, 262145, F64, 9) (dual of [262145, 262120, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(6425, 262145, F64, 9) (dual of [262145, 262120, 10]-code), using
(25−9, 25, 131073)-Net over F64 — Digital
Digital (16, 25, 131073)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6425, 131073, F64, 2, 9) (dual of [(131073, 2), 262121, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6425, 262146, F64, 9) (dual of [262146, 262121, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(6425, 262147, F64, 9) (dual of [262147, 262122, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(6425, 262144, F64, 9) (dual of [262144, 262119, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(6422, 262144, F64, 8) (dual of [262144, 262122, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(6425, 262147, F64, 9) (dual of [262147, 262122, 10]-code), using
- OOA 2-folding [i] based on linear OA(6425, 262146, F64, 9) (dual of [262146, 262121, 10]-code), using
(25−9, 25, large)-Net in Base 64 — Upper bound on s
There is no (16, 25, large)-net in base 64, because
- 7 times m-reduction [i] would yield (16, 18, large)-net in base 64, but