Best Known (89−28, 89, s)-Nets in Base 64
(89−28, 89, 18726)-Net over F64 — Constructive and digital
Digital (61, 89, 18726)-net over F64, using
- 1 times m-reduction [i] based on digital (61, 90, 18726)-net over F64, using
- net defined by OOA [i] based on linear OOA(6490, 18726, F64, 29, 29) (dual of [(18726, 29), 542964, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(6490, 262165, F64, 29) (dual of [262165, 262075, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(6490, 262168, F64, 29) (dual of [262168, 262078, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,11]) [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(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(645, 23, F64, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to C([0,14]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6490, 262168, F64, 29) (dual of [262168, 262078, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(6490, 262165, F64, 29) (dual of [262165, 262075, 30]-code), using
- net defined by OOA [i] based on linear OOA(6490, 18726, F64, 29, 29) (dual of [(18726, 29), 542964, 30]-NRT-code), using
(89−28, 89, 217346)-Net over F64 — Digital
Digital (61, 89, 217346)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6489, 217346, F64, 28) (dual of [217346, 217257, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(6489, 262175, F64, 28) (dual of [262175, 262086, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [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(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(647, 31, F64, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,64)), using
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- Reed–Solomon code RS(57,64) [i]
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(6489, 262175, F64, 28) (dual of [262175, 262086, 29]-code), using
(89−28, 89, large)-Net in Base 64 — Upper bound on s
There is no (61, 89, large)-net in base 64, because
- 26 times m-reduction [i] would yield (61, 63, large)-net in base 64, but