Best Known (58, 58+28, s)-Nets in Base 64
(58, 58+28, 18725)-Net over F64 — Constructive and digital
Digital (58, 86, 18725)-net over F64, using
- t-expansion [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
(58, 58+28, 134503)-Net over F64 — Digital
Digital (58, 86, 134503)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6486, 134503, F64, 28) (dual of [134503, 134417, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(6486, 262163, F64, 28) (dual of [262163, 262077, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(6482, 262144, F64, 28) (dual of [262144, 262062, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(6467, 262144, F64, 23) (dual of [262144, 262077, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(27) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(6486, 262163, F64, 28) (dual of [262163, 262077, 29]-code), using
(58, 58+28, large)-Net in Base 64 — Upper bound on s
There is no (58, 86, large)-net in base 64, because
- 26 times m-reduction [i] would yield (58, 60, large)-net in base 64, but