Best Known (26−13, 26, s)-Nets in Base 64
(26−13, 26, 683)-Net over F64 — Constructive and digital
Digital (13, 26, 683)-net over F64, using
- net defined by OOA [i] based on linear OOA(6426, 683, F64, 13, 13) (dual of [(683, 13), 8853, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6426, 4099, F64, 13) (dual of [4099, 4073, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6426, 4102, F64, 13) (dual of [4102, 4076, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- 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(6421, 4097, F64, 11) (dual of [4097, 4076, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6426, 4102, F64, 13) (dual of [4102, 4076, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6426, 4099, F64, 13) (dual of [4099, 4073, 14]-code), using
(26−13, 26, 1550)-Net over F64 — Digital
Digital (13, 26, 1550)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6426, 1550, F64, 2, 13) (dual of [(1550, 2), 3074, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6426, 2051, F64, 2, 13) (dual of [(2051, 2), 4076, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6426, 4102, F64, 13) (dual of [4102, 4076, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- 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(6421, 4097, F64, 11) (dual of [4097, 4076, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- OOA 2-folding [i] based on linear OA(6426, 4102, F64, 13) (dual of [4102, 4076, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(6426, 2051, F64, 2, 13) (dual of [(2051, 2), 4076, 14]-NRT-code), using
(26−13, 26, 1594522)-Net in Base 64 — Upper bound on s
There is no (13, 26, 1594523)-net in base 64, because
- 1 times m-reduction [i] would yield (13, 25, 1594523)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 1427 249073 402239 931645 173456 520574 971811 130230 > 6425 [i]