Best Known (7, 14, s)-Nets in Base 64
(7, 14, 1367)-Net over F64 — Constructive and digital
Digital (7, 14, 1367)-net over F64, using
- net defined by OOA [i] based on linear OOA(6414, 1367, F64, 7, 7) (dual of [(1367, 7), 9555, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(6414, 4102, F64, 7) (dual of [4102, 4088, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(6413, 4097, F64, 7) (dual of [4097, 4084, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(649, 4097, F64, 5) (dual of [4097, 4088, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (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([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(6414, 4102, F64, 7) (dual of [4102, 4088, 8]-code), using
(7, 14, 2052)-Net over F64 — Digital
Digital (7, 14, 2052)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6414, 2052, F64, 7) (dual of [2052, 2038, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(6414, 4102, F64, 7) (dual of [4102, 4088, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(6413, 4097, F64, 7) (dual of [4097, 4084, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(649, 4097, F64, 5) (dual of [4097, 4088, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (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([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6414, 4102, F64, 7) (dual of [4102, 4088, 8]-code), using
(7, 14, 1935632)-Net in Base 64 — Upper bound on s
There is no (7, 14, 1935633)-net in base 64, because
- 1 times m-reduction [i] would yield (7, 13, 1935633)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 302231 798889 663047 657680 > 6413 [i]