Best Known (21, 21+11, s)-Nets in Base 64
(21, 21+11, 52430)-Net over F64 — Constructive and digital
Digital (21, 32, 52430)-net over F64, using
- net defined by OOA [i] based on linear OOA(6432, 52430, F64, 11, 11) (dual of [(52430, 11), 576698, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6432, 262151, F64, 11) (dual of [262151, 262119, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6432, 262152, F64, 11) (dual of [262152, 262120, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 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(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(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([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6432, 262152, F64, 11) (dual of [262152, 262120, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6432, 262151, F64, 11) (dual of [262151, 262119, 12]-code), using
(21, 21+11, 131076)-Net over F64 — Digital
Digital (21, 32, 131076)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6432, 131076, F64, 2, 11) (dual of [(131076, 2), 262120, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6432, 262152, F64, 11) (dual of [262152, 262120, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 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(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(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([0,5]) ⊂ C([0,4]) [i] based on
- OOA 2-folding [i] based on linear OA(6432, 262152, F64, 11) (dual of [262152, 262120, 12]-code), using
(21, 21+11, large)-Net in Base 64 — Upper bound on s
There is no (21, 32, large)-net in base 64, because
- 9 times m-reduction [i] would yield (21, 23, large)-net in base 64, but