Best Known (12, 19, s)-Nets in Base 64
(12, 19, 87382)-Net over F64 — Constructive and digital
Digital (12, 19, 87382)-net over F64, using
- net defined by OOA [i] based on linear OOA(6419, 87382, F64, 7, 7) (dual of [(87382, 7), 611655, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(6419, 262147, F64, 7) (dual of [262147, 262128, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(6419, 262144, F64, 7) (dual of [262144, 262125, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(6416, 262144, F64, 6) (dual of [262144, 262128, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(6419, 262147, F64, 7) (dual of [262147, 262128, 8]-code), using
(12, 19, 131443)-Net over F64 — Digital
Digital (12, 19, 131443)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6419, 131443, F64, 7) (dual of [131443, 131424, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(6419, 262144, F64, 7) (dual of [262144, 262125, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(6419, 262144, F64, 7) (dual of [262144, 262125, 8]-code), using
(12, 19, large)-Net in Base 64 — Upper bound on s
There is no (12, 19, large)-net in base 64, because
- 5 times m-reduction [i] would yield (12, 14, large)-net in base 64, but