Best Known (54−27, 54, s)-Nets in Base 64
(54−27, 54, 354)-Net over F64 — Constructive and digital
Digital (27, 54, 354)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- digital (7, 34, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64 (see above)
- digital (7, 20, 177)-net over F64, using
(54−27, 54, 514)-Net in Base 64 — Constructive
(27, 54, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (5, 18, 257)-net in base 64, using
- 2 times m-reduction [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
- 2 times m-reduction [i] based on (5, 20, 257)-net in base 64, using
- (9, 36, 257)-net in base 64, using
- base change [i] based on digital (0, 27, 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, 27, 257)-net over F256, using
- (5, 18, 257)-net in base 64, using
(54−27, 54, 1367)-Net over F64 — Digital
Digital (27, 54, 1367)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6454, 1367, F64, 3, 27) (dual of [(1367, 3), 4047, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6454, 4101, F64, 27) (dual of [4101, 4047, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(6454, 4102, F64, 27) (dual of [4102, 4048, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(6453, 4097, F64, 27) (dual of [4097, 4044, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(6449, 4097, F64, 25) (dual of [4097, 4048, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6454, 4102, F64, 27) (dual of [4102, 4048, 28]-code), using
- OOA 3-folding [i] based on linear OA(6454, 4101, F64, 27) (dual of [4101, 4047, 28]-code), using
(54−27, 54, 2078361)-Net in Base 64 — Upper bound on s
There is no (27, 54, 2078362)-net in base 64, because
- 1 times m-reduction [i] would yield (27, 53, 2078362)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 533997 469596 310360 708138 509578 584176 816529 759223 512921 429973 681861 190855 899185 105480 274570 612404 > 6453 [i]