Best Known (8, 8+7, s)-Nets in Base 64
(8, 8+7, 1367)-Net over F64 — Constructive and digital
Digital (8, 15, 1367)-net over F64, using
- 641 times duplication [i] based on digital (7, 14, 1367)-net over F64, using
- net defined by OOA [i] based on linear OOA(6414, 1367, F64, 7, 7) (dual of [(1367, 7), 9555, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(6414, 4102, F64, 7) (dual of [4102, 4088, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(6413, 4097, F64, 7) (dual of [4097, 4084, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(649, 4097, F64, 5) (dual of [4097, 4088, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [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 C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(6414, 4102, F64, 7) (dual of [4102, 4088, 8]-code), using
- net defined by OOA [i] based on linear OOA(6414, 1367, F64, 7, 7) (dual of [(1367, 7), 9555, 8]-NRT-code), using
(8, 8+7, 4136)-Net over F64 — Digital
Digital (8, 15, 4136)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6415, 4136, F64, 7) (dual of [4136, 4121, 8]-code), using
- 36 step Varšamov–Edel lengthening with (ri) = (2, 35 times 0) [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
- 36 step Varšamov–Edel lengthening with (ri) = (2, 35 times 0) [i] based on linear OA(6413, 4098, F64, 7) (dual of [4098, 4085, 8]-code), using
(8, 8+7, 7742532)-Net in Base 64 — Upper bound on s
There is no (8, 15, 7742533)-net in base 64, because
- 1 times m-reduction [i] would yield (8, 14, 7742533)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 19 342819 068733 242403 381180 > 6414 [i]