Best Known (56−25, 56, s)-Nets in Base 64
(56−25, 56, 513)-Net over F64 — Constructive and digital
Digital (31, 56, 513)-net over F64, using
- t-expansion [i] based on digital (28, 56, 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
(56−25, 56, 545)-Net in Base 64 — Constructive
(31, 56, 545)-net in base 64, using
- (u, u+v)-construction [i] based on
- (4, 16, 257)-net in base 64, using
- base change [i] based on digital (0, 12, 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, 12, 257)-net over F256, using
- (15, 40, 288)-net in base 64, using
- 2 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
- base change [i] based on digital (9, 36, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 36, 288)-net over F128, using
- 2 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
- (4, 16, 257)-net in base 64, using
(56−25, 56, 3110)-Net over F64 — Digital
Digital (31, 56, 3110)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6456, 3110, F64, 25) (dual of [3110, 3054, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(6456, 4120, F64, 25) (dual of [4120, 4064, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- 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(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(647, 23, F64, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,64)), using
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- Reed–Solomon code RS(57,64) [i]
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6456, 4120, F64, 25) (dual of [4120, 4064, 26]-code), using
(56−25, 56, large)-Net in Base 64 — Upper bound on s
There is no (31, 56, large)-net in base 64, because
- 23 times m-reduction [i] would yield (31, 33, large)-net in base 64, but