Best Known (10, 10+8, s)-Nets in Base 64
(10, 10+8, 1026)-Net over F64 — Constructive and digital
Digital (10, 18, 1026)-net over F64, using
- 641 times duplication [i] based on digital (9, 17, 1026)-net over F64, using
- net defined by OOA [i] based on linear OOA(6417, 1026, F64, 8, 8) (dual of [(1026, 8), 8191, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(6417, 4104, F64, 8) (dual of [4104, 4087, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- 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(649, 4096, F64, 5) (dual of [4096, 4087, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(7) ⊂ Ce(4) [i] based on
- OA 4-folding and stacking [i] based on linear OA(6417, 4104, F64, 8) (dual of [4104, 4087, 9]-code), using
- net defined by OOA [i] based on linear OOA(6417, 1026, F64, 8, 8) (dual of [(1026, 8), 8191, 9]-NRT-code), using
(10, 10+8, 4096)-Net in Base 64 — Constructive
(10, 18, 4096)-net in base 64, using
- net defined by OOA [i] based on OOA(6418, 4096, S64, 8, 8), using
- OA 4-folding and stacking [i] based on OA(6418, 16384, S64, 8), using
- discarding factors based on OA(6418, 16386, S64, 8), using
- discarding parts of the base [i] based on linear OA(12815, 16386, F128, 8) (dual of [16386, 16371, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding parts of the base [i] based on linear OA(12815, 16386, F128, 8) (dual of [16386, 16371, 9]-code), using
- discarding factors based on OA(6418, 16386, S64, 8), using
- OA 4-folding and stacking [i] based on OA(6418, 16384, S64, 8), using
(10, 10+8, 4225)-Net over F64 — Digital
Digital (10, 18, 4225)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6418, 4225, F64, 8) (dual of [4225, 4207, 9]-code), using
- 124 step Varšamov–Edel lengthening with (ri) = (2, 12 times 0, 1, 110 times 0) [i] based on linear OA(6415, 4098, F64, 8) (dual of [4098, 4083, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- 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(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(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(7) ⊂ Ce(6) [i] based on
- 124 step Varšamov–Edel lengthening with (ri) = (2, 12 times 0, 1, 110 times 0) [i] based on linear OA(6415, 4098, F64, 8) (dual of [4098, 4083, 9]-code), using
(10, 10+8, 4715437)-Net in Base 64 — Upper bound on s
There is no (10, 18, 4715438)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 324 518670 492721 465835 040877 957411 > 6418 [i]