Best Known (25, 25+13, s)-Nets in Base 64
(25, 25+13, 43691)-Net over F64 — Constructive and digital
Digital (25, 38, 43691)-net over F64, using
- 641 times duplication [i] based on digital (24, 37, 43691)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 43691, F64, 13, 13) (dual of [(43691, 13), 567946, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [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(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(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- net defined by OOA [i] based on linear OOA(6437, 43691, F64, 13, 13) (dual of [(43691, 13), 567946, 14]-NRT-code), using
(25, 25+13, 131076)-Net over F64 — Digital
Digital (25, 38, 131076)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6438, 131076, F64, 2, 13) (dual of [(131076, 2), 262114, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6438, 262152, F64, 13) (dual of [262152, 262114, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [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(6431, 262145, F64, 11) (dual of [262145, 262114, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- OOA 2-folding [i] based on linear OA(6438, 262152, F64, 13) (dual of [262152, 262114, 14]-code), using
(25, 25+13, large)-Net in Base 64 — Upper bound on s
There is no (25, 38, large)-net in base 64, because
- 11 times m-reduction [i] would yield (25, 27, large)-net in base 64, but