Best Known (68−31, 68, s)-Nets in Base 64
(68−31, 68, 513)-Net over F64 — Constructive and digital
Digital (37, 68, 513)-net over F64, using
- t-expansion [i] based on digital (28, 68, 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
(68−31, 68, 545)-Net in Base 64 — Constructive
(37, 68, 545)-net in base 64, using
- 641 times duplication [i] based on (36, 67, 545)-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
- (16, 47, 288)-net in base 64, using
- 2 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- base change [i] based on digital (9, 42, 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, 42, 288)-net over F128, using
- 2 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- (5, 20, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
(68−31, 68, 2745)-Net over F64 — Digital
Digital (37, 68, 2745)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6468, 2745, F64, 31) (dual of [2745, 2677, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(6468, 4120, F64, 31) (dual of [4120, 4052, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,11]) [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(6445, 4097, F64, 23) (dual of [4097, 4052, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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([0,15]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6468, 4120, F64, 31) (dual of [4120, 4052, 32]-code), using
(68−31, 68, large)-Net in Base 64 — Upper bound on s
There is no (37, 68, large)-net in base 64, because
- 29 times m-reduction [i] would yield (37, 39, large)-net in base 64, but