Best Known (58, 58+27, s)-Nets in Base 64
(58, 58+27, 20166)-Net over F64 — Constructive and digital
Digital (58, 85, 20166)-net over F64, using
- 643 times duplication [i] based on digital (55, 82, 20166)-net over F64, using
- net defined by OOA [i] based on linear OOA(6482, 20166, F64, 27, 27) (dual of [(20166, 27), 544400, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(6482, 262159, F64, 27) (dual of [262159, 262077, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(6482, 262160, F64, 27) (dual of [262160, 262078, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- 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(6467, 262145, F64, 23) (dual of [262145, 262078, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(643, 15, F64, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,64) or 15-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6482, 262160, F64, 27) (dual of [262160, 262078, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(6482, 262159, F64, 27) (dual of [262159, 262077, 28]-code), using
- net defined by OOA [i] based on linear OOA(6482, 20166, F64, 27, 27) (dual of [(20166, 27), 544400, 28]-NRT-code), using
(58, 58+27, 189244)-Net over F64 — Digital
Digital (58, 85, 189244)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6485, 189244, F64, 27) (dual of [189244, 189159, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(6485, 262171, F64, 27) (dual of [262171, 262086, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(19) [i] based on
- linear OA(6479, 262144, F64, 27) (dual of [262144, 262065, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(646, 27, F64, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,64)), using
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- Reed–Solomon code RS(58,64) [i]
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- construction X applied to Ce(26) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(6485, 262171, F64, 27) (dual of [262171, 262086, 28]-code), using
(58, 58+27, large)-Net in Base 64 — Upper bound on s
There is no (58, 85, large)-net in base 64, because
- 25 times m-reduction [i] would yield (58, 60, large)-net in base 64, but