Best Known (48, 48+42, s)-Nets in Base 64
(48, 48+42, 513)-Net over F64 — Constructive and digital
Digital (48, 90, 513)-net over F64, using
- t-expansion [i] based on digital (28, 90, 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
(48, 48+42, 546)-Net in Base 64 — Constructive
(48, 90, 546)-net in base 64, using
- (u, u+v)-construction [i] based on
- (9, 30, 258)-net in base 64, using
- 2 times m-reduction [i] based on (9, 32, 258)-net in base 64, using
- base change [i] based on digital (1, 24, 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, 24, 258)-net over F256, using
- 2 times m-reduction [i] based on (9, 32, 258)-net in base 64, using
- (18, 60, 288)-net in base 64, using
- 3 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
- base change [i] based on digital (9, 54, 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, 54, 288)-net over F128, using
- 3 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
- (9, 30, 258)-net in base 64, using
(48, 48+42, 2594)-Net over F64 — Digital
Digital (48, 90, 2594)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6490, 2594, F64, 42) (dual of [2594, 2504, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(6490, 4119, F64, 42) (dual of [4119, 4029, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(33) [i] based on
- linear OA(6483, 4096, F64, 42) (dual of [4096, 4013, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(6467, 4096, F64, 34) (dual of [4096, 4029, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [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 Ce(41) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(6490, 4119, F64, 42) (dual of [4119, 4029, 43]-code), using
(48, 48+42, 7584426)-Net in Base 64 — Upper bound on s
There is no (48, 90, 7584427)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 3 599136 686900 200161 674414 468041 697826 167902 222116 121169 360381 258052 538011 949089 155731 038745 716857 963117 986000 536358 442771 590567 459132 637386 237625 733982 102009 196632 > 6490 [i]