Best Known (41−13, 41, s)-Nets in Base 64
(41−13, 41, 43693)-Net over F64 — Constructive and digital
Digital (28, 41, 43693)-net over F64, using
- 641 times duplication [i] based on digital (27, 40, 43693)-net over F64, using
- net defined by OOA [i] based on linear OOA(6440, 43693, F64, 13, 13) (dual of [(43693, 13), 567969, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6440, 262159, F64, 13) (dual of [262159, 262119, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6440, 262160, F64, 13) (dual of [262160, 262120, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- 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(6425, 262145, F64, 9) (dual of [262145, 262120, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6440, 262160, F64, 13) (dual of [262160, 262120, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6440, 262159, F64, 13) (dual of [262159, 262119, 14]-code), using
- net defined by OOA [i] based on linear OOA(6440, 43693, F64, 13, 13) (dual of [(43693, 13), 567969, 14]-NRT-code), using
(41−13, 41, 262163)-Net over F64 — Digital
Digital (28, 41, 262163)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6441, 262163, F64, 13) (dual of [262163, 262122, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(6422, 262144, F64, 8) (dual of [262144, 262122, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(644, 19, F64, 4) (dual of [19, 15, 5]-code or 19-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(12) ⊂ Ce(7) [i] based on
(41−13, 41, large)-Net in Base 64 — Upper bound on s
There is no (28, 41, large)-net in base 64, because
- 11 times m-reduction [i] would yield (28, 30, large)-net in base 64, but