Best Known (14, 14+13, s)-Nets in Base 64
(14, 14+13, 683)-Net over F64 — Constructive and digital
Digital (14, 27, 683)-net over F64, using
- 641 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(6426, 683, F64, 13, 13) (dual of [(683, 13), 8853, 14]-NRT-code), using
(14, 14+13, 2052)-Net over F64 — Digital
Digital (14, 27, 2052)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6427, 2052, F64, 2, 13) (dual of [(2052, 2), 4077, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6427, 4104, F64, 13) (dual of [4104, 4077, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(6425, 4096, F64, 13) (dual of [4096, 4071, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(6427, 4104, F64, 13) (dual of [4104, 4077, 14]-code), using
(14, 14+13, 3189048)-Net in Base 64 — Upper bound on s
There is no (14, 27, 3189049)-net in base 64, because
- 1 times m-reduction [i] would yield (14, 26, 3189049)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 91344 010259 571709 742783 200211 597617 312347 311188 > 6426 [i]