Best Known (13−7, 13, s)-Nets in Base 64
(13−7, 13, 1365)-Net over F64 — Constructive and digital
Digital (6, 13, 1365)-net over F64, using
- net defined by OOA [i] based on linear OOA(6413, 1365, F64, 7, 7) (dual of [(1365, 7), 9542, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using
(13−7, 13, 2049)-Net over F64 — Digital
Digital (6, 13, 2049)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6413, 2049, F64, 2, 7) (dual of [(2049, 2), 4085, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6413, 4098, F64, 7) (dual of [4098, 4085, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(6411, 4096, F64, 6) (dual of [4096, 4085, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- OOA 2-folding [i] based on linear OA(6413, 4098, F64, 7) (dual of [4098, 4085, 8]-code), using
(13−7, 13, 483907)-Net in Base 64 — Upper bound on s
There is no (6, 13, 483908)-net in base 64, because
- 1 times m-reduction [i] would yield (6, 12, 483908)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 4722 387541 450584 633055 > 6412 [i]