Best Known (80−37, 80, s)-Nets in Base 64
(80−37, 80, 513)-Net over F64 — Constructive and digital
Digital (43, 80, 513)-net over F64, using
- t-expansion [i] based on digital (28, 80, 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
(80−37, 80, 546)-Net in Base 64 — Constructive
(43, 80, 546)-net in base 64, using
- (u, u+v)-construction [i] based on
- (8, 26, 258)-net in base 64, using
- 2 times m-reduction [i] based on (8, 28, 258)-net in base 64, using
- base change [i] based on digital (1, 21, 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, 21, 258)-net over F256, using
- 2 times m-reduction [i] based on (8, 28, 258)-net in base 64, using
- (17, 54, 288)-net in base 64, using
- 2 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 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, 48, 288)-net over F128, using
- 2 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- (8, 26, 258)-net in base 64, using
(80−37, 80, 2618)-Net over F64 — Digital
Digital (43, 80, 2618)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6480, 2618, F64, 37) (dual of [2618, 2538, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(6480, 4120, F64, 37) (dual of [4120, 4040, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,14]) [i] based on
- linear OA(6473, 4097, F64, 37) (dual of [4097, 4024, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (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(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([0,18]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6480, 4120, F64, 37) (dual of [4120, 4040, 38]-code), using
(80−37, 80, large)-Net in Base 64 — Upper bound on s
There is no (43, 80, large)-net in base 64, because
- 35 times m-reduction [i] would yield (43, 45, large)-net in base 64, but