Best Known (31−10, 31, s)-Nets in Base 64
(31−10, 31, 52431)-Net over F64 — Constructive and digital
Digital (21, 31, 52431)-net over F64, using
- 641 times duplication [i] based on digital (20, 30, 52431)-net over F64, using
- net defined by OOA [i] based on linear OOA(6430, 52431, F64, 10, 10) (dual of [(52431, 10), 524280, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6430, 262155, F64, 10) (dual of [262155, 262125, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(6419, 262144, F64, 7) (dual of [262144, 262125, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(642, 11, F64, 2) (dual of [11, 9, 3]-code or 11-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(9) ⊂ Ce(6) [i] based on
- OA 5-folding and stacking [i] based on linear OA(6430, 262155, F64, 10) (dual of [262155, 262125, 11]-code), using
- net defined by OOA [i] based on linear OOA(6430, 52431, F64, 10, 10) (dual of [(52431, 10), 524280, 11]-NRT-code), using
(31−10, 31, 262159)-Net over F64 — Digital
Digital (21, 31, 262159)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6431, 262159, F64, 10) (dual of [262159, 262128, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(6416, 262144, F64, 6) (dual of [262144, 262128, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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 Ce(9) ⊂ Ce(5) [i] based on
(31−10, 31, large)-Net in Base 64 — Upper bound on s
There is no (21, 31, large)-net in base 64, because
- 8 times m-reduction [i] would yield (21, 23, large)-net in base 64, but