Best Known (78−39, 78, s)-Nets in Base 64
(78−39, 78, 513)-Net over F64 — Constructive and digital
Digital (39, 78, 513)-net over F64, using
- t-expansion [i] based on digital (28, 78, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
(78−39, 78, 514)-Net in Base 64 — Constructive
(39, 78, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (7, 26, 257)-net in base 64, using
- 2 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
- base change [i] based on digital (0, 21, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 21, 257)-net over F256, using
- 2 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
- (13, 52, 257)-net in base 64, using
- base change [i] based on digital (0, 39, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- base change [i] based on digital (0, 39, 257)-net over F256, using
- (7, 26, 257)-net in base 64, using
(78−39, 78, 1456)-Net over F64 — Digital
Digital (39, 78, 1456)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6478, 1456, F64, 2, 39) (dual of [(1456, 2), 2834, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6478, 2051, F64, 2, 39) (dual of [(2051, 2), 4024, 40]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6478, 4102, F64, 39) (dual of [4102, 4024, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- linear OA(6477, 4097, F64, 39) (dual of [4097, 4020, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(6473, 4097, F64, 37) (dual of [4097, 4024, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (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([0,19]) ⊂ C([0,18]) [i] based on
- OOA 2-folding [i] based on linear OA(6478, 4102, F64, 39) (dual of [4102, 4024, 40]-code), using
- discarding factors / shortening the dual code based on linear OOA(6478, 2051, F64, 2, 39) (dual of [(2051, 2), 4024, 40]-NRT-code), using
(78−39, 78, 2628181)-Net in Base 64 — Upper bound on s
There is no (39, 78, 2628182)-net in base 64, because
- 1 times m-reduction [i] would yield (39, 77, 2628182)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 11 908534 432492 451912 313992 522678 925554 272038 878804 987017 713251 835788 185483 679618 015048 641397 905486 725702 651457 114709 499573 704938 928915 628982 > 6477 [i]