Best Known (26−9, 26, s)-Nets in Base 64
(26−9, 26, 65537)-Net over F64 — Constructive and digital
Digital (17, 26, 65537)-net over F64, using
- net defined by OOA [i] based on linear OOA(6426, 65537, F64, 9, 9) (dual of [(65537, 9), 589807, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6426, 262149, F64, 9) (dual of [262149, 262123, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(6426, 262152, F64, 9) (dual of [262152, 262126, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- 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(6419, 262145, F64, 7) (dual of [262145, 262126, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6426, 262152, F64, 9) (dual of [262152, 262126, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6426, 262149, F64, 9) (dual of [262149, 262123, 10]-code), using
(26−9, 26, 151431)-Net over F64 — Digital
Digital (17, 26, 151431)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6426, 151431, F64, 9) (dual of [151431, 151405, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(6426, 262152, F64, 9) (dual of [262152, 262126, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- 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(6419, 262145, F64, 7) (dual of [262145, 262126, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6426, 262152, F64, 9) (dual of [262152, 262126, 10]-code), using
(26−9, 26, large)-Net in Base 64 — Upper bound on s
There is no (17, 26, large)-net in base 64, because
- 7 times m-reduction [i] would yield (17, 19, large)-net in base 64, but