Best Known (6, 6+6, s)-Nets in Base 64
(6, 6+6, 1367)-Net over F64 — Constructive and digital
Digital (6, 12, 1367)-net over F64, using
- net defined by OOA [i] based on linear OOA(6412, 1367, F64, 6, 6) (dual of [(1367, 6), 8190, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(6412, 4101, F64, 6) (dual of [4101, 4089, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- 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(647, 4096, F64, 4) (dual of [4096, 4089, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(6412, 4101, F64, 6) (dual of [4101, 4089, 7]-code), using
(6, 6+6, 3255)-Net over F64 — Digital
Digital (6, 12, 3255)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6412, 3255, F64, 6) (dual of [3255, 3243, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(6412, 4101, F64, 6) (dual of [4101, 4089, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- 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(647, 4096, F64, 4) (dual of [4096, 4089, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(5) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(6412, 4101, F64, 6) (dual of [4101, 4089, 7]-code), using
(6, 6+6, 483907)-Net in Base 64 — Upper bound on s
There is no (6, 12, 483908)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 4722 387541 450584 633055 > 6412 [i]