Best Known (48, 48+43, s)-Nets in Base 64
(48, 48+43, 513)-Net over F64 — Constructive and digital
Digital (48, 91, 513)-net over F64, using
- t-expansion [i] based on digital (28, 91, 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+43, 546)-Net in Base 64 — Constructive
(48, 91, 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, 61, 288)-net in base 64, using
- 2 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
- 2 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
- (9, 30, 258)-net in base 64, using
(48, 48+43, 2343)-Net over F64 — Digital
Digital (48, 91, 2343)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6491, 2343, F64, 43) (dual of [2343, 2252, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(6491, 4116, F64, 43) (dual of [4116, 4025, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(35) [i] based on
- linear OA(6485, 4096, F64, 43) (dual of [4096, 4011, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(6471, 4096, F64, 36) (dual of [4096, 4025, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(646, 20, F64, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,64)), using
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- Reed–Solomon code RS(58,64) [i]
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- construction X applied to Ce(42) ⊂ Ce(35) [i] based on
- discarding factors / shortening the dual code based on linear OA(6491, 4116, F64, 43) (dual of [4116, 4025, 44]-code), using
(48, 48+43, 7584426)-Net in Base 64 — Upper bound on s
There is no (48, 91, 7584427)-net in base 64, because
- 1 times m-reduction [i] would yield (48, 90, 7584427)-net in base 64, but
- 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]