Best Known (30, 45, s)-Nets in Base 64
(30, 45, 37450)-Net over F64 — Constructive and digital
Digital (30, 45, 37450)-net over F64, using
- 641 times duplication [i] based on digital (29, 44, 37450)-net over F64, using
- net defined by OOA [i] based on linear OOA(6444, 37450, F64, 15, 15) (dual of [(37450, 15), 561706, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6444, 262151, F64, 15) (dual of [262151, 262107, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6444, 262152, F64, 15) (dual of [262152, 262108, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(6443, 262145, F64, 15) (dual of [262145, 262102, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(6437, 262145, F64, 13) (dual of [262145, 262108, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6444, 262152, F64, 15) (dual of [262152, 262108, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6444, 262151, F64, 15) (dual of [262151, 262107, 16]-code), using
- net defined by OOA [i] based on linear OOA(6444, 37450, F64, 15, 15) (dual of [(37450, 15), 561706, 16]-NRT-code), using
(30, 45, 131077)-Net over F64 — Digital
Digital (30, 45, 131077)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6445, 131077, F64, 2, 15) (dual of [(131077, 2), 262109, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6445, 262154, F64, 15) (dual of [262154, 262109, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6445, 262155, F64, 15) (dual of [262155, 262110, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(6443, 262144, F64, 15) (dual of [262144, 262101, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(6434, 262144, F64, 12) (dual of [262144, 262110, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(6445, 262155, F64, 15) (dual of [262155, 262110, 16]-code), using
- OOA 2-folding [i] based on linear OA(6445, 262154, F64, 15) (dual of [262154, 262109, 16]-code), using
(30, 45, large)-Net in Base 64 — Upper bound on s
There is no (30, 45, large)-net in base 64, because
- 13 times m-reduction [i] would yield (30, 32, large)-net in base 64, but