Best Known (21−11, 21, s)-Nets in Base 64
(21−11, 21, 819)-Net over F64 — Constructive and digital
Digital (10, 21, 819)-net over F64, using
- net defined by OOA [i] based on linear OOA(6421, 819, F64, 11, 11) (dual of [(819, 11), 8988, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- OOA 5-folding and stacking with additional row [i] based on linear OA(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using
(21−11, 21, 1366)-Net over F64 — Digital
Digital (10, 21, 1366)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6421, 1366, F64, 3, 11) (dual of [(1366, 3), 4077, 12]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6421, 4098, F64, 11) (dual of [4098, 4077, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(640, 2, F64, 0) (dual of [2, 2, 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
- OOA 3-folding [i] based on linear OA(6421, 4098, F64, 11) (dual of [4098, 4077, 12]-code), using
(21−11, 21, 693768)-Net in Base 64 — Upper bound on s
There is no (10, 21, 693769)-net in base 64, because
- 1 times m-reduction [i] would yield (10, 20, 693769)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 1 329237 118610 918966 992809 967689 316168 > 6420 [i]