Best Known (15, 15+15, s)-Nets in Base 64
(15, 15+15, 585)-Net over F64 — Constructive and digital
Digital (15, 30, 585)-net over F64, using
- 641 times duplication [i] based on digital (14, 29, 585)-net over F64, using
- net defined by OOA [i] based on linear OOA(6429, 585, F64, 15, 15) (dual of [(585, 15), 8746, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using
- net defined by OOA [i] based on linear OOA(6429, 585, F64, 15, 15) (dual of [(585, 15), 8746, 16]-NRT-code), using
(15, 15+15, 1370)-Net over F64 — Digital
Digital (15, 30, 1370)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6430, 1370, F64, 2, 15) (dual of [(1370, 2), 2710, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6430, 2051, F64, 2, 15) (dual of [(2051, 2), 4072, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6430, 4102, F64, 15) (dual of [4102, 4072, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(6429, 4097, F64, 15) (dual of [4097, 4068, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 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([0,7]) ⊂ C([0,6]) [i] based on
- OOA 2-folding [i] based on linear OA(6430, 4102, F64, 15) (dual of [4102, 4072, 16]-code), using
- discarding factors / shortening the dual code based on linear OOA(6430, 2051, F64, 2, 15) (dual of [(2051, 2), 4072, 16]-NRT-code), using
(15, 15+15, 1630507)-Net in Base 64 — Upper bound on s
There is no (15, 30, 1630508)-net in base 64, because
- 1 times m-reduction [i] would yield (15, 29, 1630508)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 23945 265031 716989 142721 865725 193106 153877 273435 149646 > 6429 [i]