Best Known (31, 31+31, s)-Nets in Base 64
(31, 31+31, 513)-Net over F64 — Constructive and digital
Digital (31, 62, 513)-net over F64, using
- t-expansion [i] based on digital (28, 62, 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
(31, 31+31, 514)-Net in Base 64 — Constructive
(31, 62, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (5, 20, 257)-net in base 64, using
- base change [i] based on digital (0, 15, 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, 15, 257)-net over F256, using
- (11, 42, 257)-net in base 64, using
- 2 times m-reduction [i] based on (11, 44, 257)-net in base 64, using
- base change [i] based on digital (0, 33, 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, 33, 257)-net over F256, using
- 2 times m-reduction [i] based on (11, 44, 257)-net in base 64, using
- (5, 20, 257)-net in base 64, using
(31, 31+31, 1367)-Net over F64 — Digital
Digital (31, 62, 1367)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6462, 1367, F64, 3, 31) (dual of [(1367, 3), 4039, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6462, 4101, F64, 31) (dual of [4101, 4039, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(6462, 4102, F64, 31) (dual of [4102, 4040, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,14]) [i] based on
- linear OA(6461, 4097, F64, 31) (dual of [4097, 4036, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(6457, 4097, F64, 29) (dual of [4097, 4040, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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([0,15]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6462, 4102, F64, 31) (dual of [4102, 4040, 32]-code), using
- OOA 3-folding [i] based on linear OA(6462, 4101, F64, 31) (dual of [4101, 4039, 32]-code), using
(31, 31+31, 2257130)-Net in Base 64 — Upper bound on s
There is no (31, 62, 2257131)-net in base 64, because
- 1 times m-reduction [i] would yield (31, 61, 2257131)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 150 307621 872892 518766 389090 326566 290759 234556 324342 569682 288925 131717 328838 211186 504283 933140 631125 870679 429728 > 6461 [i]