Best Known (20, 31, s)-Nets in Base 64
(20, 31, 52429)-Net over F64 — Constructive and digital
Digital (20, 31, 52429)-net over F64, using
- net defined by OOA [i] based on linear OOA(6431, 52429, F64, 11, 11) (dual of [(52429, 11), 576688, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6431, 262146, F64, 11) (dual of [262146, 262115, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6431, 262147, F64, 11) (dual of [262147, 262116, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(6431, 262144, F64, 11) (dual of [262144, 262113, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(6431, 262147, F64, 11) (dual of [262147, 262116, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6431, 262146, F64, 11) (dual of [262146, 262115, 12]-code), using
(20, 31, 131073)-Net over F64 — Digital
Digital (20, 31, 131073)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6431, 131073, F64, 2, 11) (dual of [(131073, 2), 262115, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6431, 262146, F64, 11) (dual of [262146, 262115, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6431, 262147, F64, 11) (dual of [262147, 262116, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(6431, 262144, F64, 11) (dual of [262144, 262113, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(6431, 262147, F64, 11) (dual of [262147, 262116, 12]-code), using
- OOA 2-folding [i] based on linear OA(6431, 262146, F64, 11) (dual of [262146, 262115, 12]-code), using
(20, 31, large)-Net in Base 64 — Upper bound on s
There is no (20, 31, large)-net in base 64, because
- 9 times m-reduction [i] would yield (20, 22, large)-net in base 64, but