Best Known (57, 57+28, s)-Nets in Base 64
(57, 57+28, 18725)-Net over F64 — Constructive and digital
Digital (57, 85, 18725)-net over F64, using
- 1 times m-reduction [i] based on digital (57, 86, 18725)-net over F64, using
- net defined by OOA [i] based on linear OOA(6486, 18725, F64, 29, 29) (dual of [(18725, 29), 542939, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(6486, 262151, F64, 29) (dual of [262151, 262065, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(6486, 262152, F64, 29) (dual of [262152, 262066, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,13]) [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(6479, 262145, F64, 27) (dual of [262145, 262066, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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([0,14]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6486, 262152, F64, 29) (dual of [262152, 262066, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(6486, 262151, F64, 29) (dual of [262151, 262065, 30]-code), using
- net defined by OOA [i] based on linear OOA(6486, 18725, F64, 29, 29) (dual of [(18725, 29), 542939, 30]-NRT-code), using
(57, 57+28, 131079)-Net over F64 — Digital
Digital (57, 85, 131079)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6485, 131079, F64, 2, 28) (dual of [(131079, 2), 262073, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6485, 262158, F64, 28) (dual of [262158, 262073, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(6485, 262159, F64, 28) (dual of [262159, 262074, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(23) [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(6470, 262144, F64, 24) (dual of [262144, 262074, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(27) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(6485, 262159, F64, 28) (dual of [262159, 262074, 29]-code), using
- OOA 2-folding [i] based on linear OA(6485, 262158, F64, 28) (dual of [262158, 262073, 29]-code), using
(57, 57+28, large)-Net in Base 64 — Upper bound on s
There is no (57, 85, large)-net in base 64, because
- 26 times m-reduction [i] would yield (57, 59, large)-net in base 64, but