Best Known (60, 60+29, s)-Nets in Base 64
(60, 60+29, 18725)-Net over F64 — Constructive and digital
Digital (60, 89, 18725)-net over F64, using
- 643 times duplication [i] based on digital (57, 86, 18725)-net over F64, using
- net defined by OOA [i] based on linear OOA(6486, 18725, F64, 29, 29) (dual of [(18725, 29), 542939, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(6486, 262151, F64, 29) (dual of [262151, 262065, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(6486, 262152, F64, 29) (dual of [262152, 262066, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,13]) [i] based on
- linear OA(6485, 262145, F64, 29) (dual of [262145, 262060, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(6479, 262145, F64, 27) (dual of [262145, 262066, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 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([0,14]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6486, 262152, F64, 29) (dual of [262152, 262066, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(6486, 262151, F64, 29) (dual of [262151, 262065, 30]-code), using
- net defined by OOA [i] based on linear OOA(6486, 18725, F64, 29, 29) (dual of [(18725, 29), 542939, 30]-NRT-code), using
(60, 60+29, 133608)-Net over F64 — Digital
Digital (60, 89, 133608)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6489, 133608, F64, 29) (dual of [133608, 133519, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(6489, 262163, F64, 29) (dual of [262163, 262074, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- linear OA(6485, 262144, F64, 29) (dual of [262144, 262059, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(6470, 262144, F64, 24) (dual of [262144, 262074, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(644, 19, F64, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(6489, 262163, F64, 29) (dual of [262163, 262074, 30]-code), using
(60, 60+29, large)-Net in Base 64 — Upper bound on s
There is no (60, 89, large)-net in base 64, because
- 27 times m-reduction [i] would yield (60, 62, large)-net in base 64, but