Best Known (10, 10+10, s)-Nets in Base 64
(10, 10+10, 820)-Net over F64 — Constructive and digital
Digital (10, 20, 820)-net over F64, using
- net defined by OOA [i] based on linear OOA(6420, 820, F64, 10, 10) (dual of [(820, 10), 8180, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6420, 4100, F64, 10) (dual of [4100, 4080, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6420, 4101, F64, 10) (dual of [4101, 4081, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(6420, 4101, F64, 10) (dual of [4101, 4081, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6420, 4100, F64, 10) (dual of [4100, 4080, 11]-code), using
(10, 10+10, 2050)-Net over F64 — Digital
Digital (10, 20, 2050)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6420, 2050, F64, 2, 10) (dual of [(2050, 2), 4080, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6420, 4100, F64, 10) (dual of [4100, 4080, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6420, 4101, F64, 10) (dual of [4101, 4081, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(6420, 4101, F64, 10) (dual of [4101, 4081, 11]-code), using
- OOA 2-folding [i] based on linear OA(6420, 4100, F64, 10) (dual of [4100, 4080, 11]-code), using
(10, 10+10, 693768)-Net in Base 64 — Upper bound on s
There is no (10, 20, 693769)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 1 329237 118610 918966 992809 967689 316168 > 6420 [i]