Best Known (17−9, 17, s)-Nets in Base 64
(17−9, 17, 1024)-Net over F64 — Constructive and digital
Digital (8, 17, 1024)-net over F64, using
- net defined by OOA [i] based on linear OOA(6417, 1024, F64, 9, 9) (dual of [(1024, 9), 9199, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using
(17−9, 17, 1555)-Net over F64 — Digital
Digital (8, 17, 1555)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6417, 1555, F64, 2, 9) (dual of [(1555, 2), 3093, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6417, 2049, F64, 2, 9) (dual of [(2049, 2), 4081, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(6417, 4096, F64, 9) (dual of [4096, 4079, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(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(8) ⊂ Ce(7) [i] based on
- OOA 2-folding [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(6417, 2049, F64, 2, 9) (dual of [(2049, 2), 4081, 10]-NRT-code), using
(17−9, 17, 589428)-Net in Base 64 — Upper bound on s
There is no (8, 17, 589429)-net in base 64, because
- 1 times m-reduction [i] would yield (8, 16, 589429)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 79228 523341 739543 082999 170869 > 6416 [i]