Best Known (19, 19+15, s)-Nets in Base 64
(19, 19+15, 587)-Net over F64 — Constructive and digital
Digital (19, 34, 587)-net over F64, using
- 641 times duplication [i] based on digital (18, 33, 587)-net over F64, using
- net defined by OOA [i] based on linear OOA(6433, 587, F64, 15, 15) (dual of [(587, 15), 8772, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6433, 4110, F64, 15) (dual of [4110, 4077, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(644, 14, F64, 4) (dual of [14, 10, 5]-code or 14-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(14) ⊂ Ce(9) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(6433, 4110, F64, 15) (dual of [4110, 4077, 16]-code), using
- net defined by OOA [i] based on linear OOA(6433, 587, F64, 15, 15) (dual of [(587, 15), 8772, 16]-NRT-code), using
(19, 19+15, 2340)-Net in Base 64 — Constructive
(19, 34, 2340)-net in base 64, using
- net defined by OOA [i] based on OOA(6434, 2340, S64, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(6434, 16381, S64, 15), using
- discarding factors based on OA(6434, 16386, S64, 15), using
- discarding parts of the base [i] based on linear OA(12829, 16386, F128, 15) (dual of [16386, 16357, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(12829, 16384, F128, 15) (dual of [16384, 16355, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(12827, 16384, F128, 14) (dual of [16384, 16357, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(12829, 16386, F128, 15) (dual of [16386, 16357, 16]-code), using
- discarding factors based on OA(6434, 16386, S64, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(6434, 16381, S64, 15), using
(19, 19+15, 3454)-Net over F64 — Digital
Digital (19, 34, 3454)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6434, 3454, F64, 15) (dual of [3454, 3420, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6434, 4114, F64, 15) (dual of [4114, 4080, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- linear OA(6429, 4097, F64, 15) (dual of [4097, 4068, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-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([0,7]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6434, 4114, F64, 15) (dual of [4114, 4080, 16]-code), using
(19, 19+15, large)-Net in Base 64 — Upper bound on s
There is no (19, 34, large)-net in base 64, because
- 13 times m-reduction [i] would yield (19, 21, large)-net in base 64, but