Best Known (72−33, 72, s)-Nets in Base 64
(72−33, 72, 513)-Net over F64 — Constructive and digital
Digital (39, 72, 513)-net over F64, using
- t-expansion [i] based on digital (28, 72, 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
(72−33, 72, 546)-Net in Base 64 — Constructive
(39, 72, 546)-net in base 64, using
- (u, u+v)-construction [i] based on
- (7, 23, 258)-net in base 64, using
- 1 times m-reduction [i] based on (7, 24, 258)-net in base 64, using
- base change [i] based on digital (1, 18, 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, 18, 258)-net over F256, using
- 1 times m-reduction [i] based on (7, 24, 258)-net in base 64, using
- (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
- (7, 23, 258)-net in base 64, using
(72−33, 72, 2686)-Net over F64 — Digital
Digital (39, 72, 2686)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6472, 2686, F64, 33) (dual of [2686, 2614, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(6472, 4120, F64, 33) (dual of [4120, 4048, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- linear OA(6465, 4097, F64, 33) (dual of [4097, 4032, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (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(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([0,16]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6472, 4120, F64, 33) (dual of [4120, 4048, 34]-code), using
(72−33, 72, large)-Net in Base 64 — Upper bound on s
There is no (39, 72, large)-net in base 64, because
- 31 times m-reduction [i] would yield (39, 41, large)-net in base 64, but