Best Known (49, 49+25, s)-Nets in Base 64
(49, 49+25, 21845)-Net over F64 — Constructive and digital
Digital (49, 74, 21845)-net over F64, using
- 641 times duplication [i] based on digital (48, 73, 21845)-net over F64, using
- net defined by OOA [i] based on linear OOA(6473, 21845, F64, 25, 25) (dual of [(21845, 25), 546052, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(6473, 262141, F64, 25) (dual of [262141, 262068, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(6473, 262144, F64, 25) (dual of [262144, 262071, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(6473, 262144, F64, 25) (dual of [262144, 262071, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(6473, 262141, F64, 25) (dual of [262141, 262068, 26]-code), using
- net defined by OOA [i] based on linear OOA(6473, 21845, F64, 25, 25) (dual of [(21845, 25), 546052, 26]-NRT-code), using
(49, 49+25, 117114)-Net over F64 — Digital
Digital (49, 74, 117114)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6474, 117114, F64, 2, 25) (dual of [(117114, 2), 234154, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6474, 131076, F64, 2, 25) (dual of [(131076, 2), 262078, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6474, 262152, F64, 25) (dual of [262152, 262078, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(6473, 262145, F64, 25) (dual of [262145, 262072, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(6467, 262145, F64, 23) (dual of [262145, 262078, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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([0,12]) ⊂ C([0,11]) [i] based on
- OOA 2-folding [i] based on linear OA(6474, 262152, F64, 25) (dual of [262152, 262078, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(6474, 131076, F64, 2, 25) (dual of [(131076, 2), 262078, 26]-NRT-code), using
(49, 49+25, large)-Net in Base 64 — Upper bound on s
There is no (49, 74, large)-net in base 64, because
- 23 times m-reduction [i] would yield (49, 51, large)-net in base 64, but