Best Known (10, 10+9, s)-Nets in Base 64
(10, 10+9, 1025)-Net over F64 — Constructive and digital
Digital (10, 19, 1025)-net over F64, using
- 641 times duplication [i] based on digital (9, 18, 1025)-net over F64, using
- net defined by OOA [i] based on linear OOA(6418, 1025, F64, 9, 9) (dual of [(1025, 9), 9207, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6418, 4101, F64, 9) (dual of [4101, 4083, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(6418, 4102, F64, 9) (dual of [4102, 4084, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [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]
- 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(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([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6418, 4102, F64, 9) (dual of [4102, 4084, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6418, 4101, F64, 9) (dual of [4101, 4083, 10]-code), using
- net defined by OOA [i] based on linear OOA(6418, 1025, F64, 9, 9) (dual of [(1025, 9), 9207, 10]-NRT-code), using
(10, 10+9, 2364)-Net over F64 — Digital
Digital (10, 19, 2364)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6419, 2364, F64, 9) (dual of [2364, 2345, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(6419, 4104, F64, 9) (dual of [4104, 4085, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [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(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(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(6419, 4104, F64, 9) (dual of [4104, 4085, 10]-code), using
(10, 10+9, 4715437)-Net in Base 64 — Upper bound on s
There is no (10, 19, 4715438)-net in base 64, because
- 1 times m-reduction [i] would yield (10, 18, 4715438)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 324 518670 492721 465835 040877 957411 > 6418 [i]