Best Known (33−17, 33, s)-Nets in Base 64
(33−17, 33, 512)-Net over F64 — Constructive and digital
Digital (16, 33, 512)-net over F64, using
- net defined by OOA [i] based on linear OOA(6433, 512, F64, 17, 17) (dual of [(512, 17), 8671, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using
(33−17, 33, 1319)-Net over F64 — Digital
Digital (16, 33, 1319)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6433, 1319, F64, 3, 17) (dual of [(1319, 3), 3924, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6433, 1366, F64, 3, 17) (dual of [(1366, 3), 4065, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6433, 4098, F64, 17) (dual of [4098, 4065, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(6433, 4096, F64, 17) (dual of [4096, 4063, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(6431, 4096, F64, 16) (dual of [4096, 4065, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(16) ⊂ Ce(15) [i] based on
- OOA 3-folding [i] based on linear OA(6433, 4098, F64, 17) (dual of [4098, 4065, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(6433, 1366, F64, 3, 17) (dual of [(1366, 3), 4065, 18]-NRT-code), using
(33−17, 33, 1002461)-Net in Base 64 — Upper bound on s
There is no (16, 33, 1002462)-net in base 64, because
- 1 times m-reduction [i] would yield (16, 32, 1002462)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 6277 110793 704836 896664 457014 278421 166965 456229 606618 532585 > 6432 [i]