Best Known (37, 37+29, s)-Nets in Base 64
(37, 37+29, 513)-Net over F64 — Constructive and digital
Digital (37, 66, 513)-net over F64, using
- t-expansion [i] based on digital (28, 66, 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
(37, 37+29, 546)-Net in Base 64 — Constructive
(37, 66, 546)-net in base 64, using
- 641 times duplication [i] based on (36, 65, 546)-net in base 64, using
- (u, u+v)-construction [i] based on
- (6, 20, 258)-net in base 64, using
- base change [i] based on digital (1, 15, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 15, 258)-net over F256, using
- (16, 45, 288)-net in base 64, using
- 4 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
- 4 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- (6, 20, 258)-net in base 64, using
- (u, u+v)-construction [i] based on
(37, 37+29, 3853)-Net over F64 — Digital
Digital (37, 66, 3853)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6466, 3853, F64, 29) (dual of [3853, 3787, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(6466, 4126, F64, 29) (dual of [4126, 4060, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,9]) [i] based on
- 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(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(649, 29, F64, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,64)), using
- discarding factors / shortening the dual code based on linear OA(649, 64, F64, 9) (dual of [64, 55, 10]-code or 64-arc in PG(8,64)), using
- Reed–Solomon code RS(55,64) [i]
- discarding factors / shortening the dual code based on linear OA(649, 64, F64, 9) (dual of [64, 55, 10]-code or 64-arc in PG(8,64)), using
- construction X applied to C([0,14]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6466, 4126, F64, 29) (dual of [4126, 4060, 30]-code), using
(37, 37+29, large)-Net in Base 64 — Upper bound on s
There is no (37, 66, large)-net in base 64, because
- 27 times m-reduction [i] would yield (37, 39, large)-net in base 64, but