Best Known (45, 45+43, s)-Nets in Base 64
(45, 45+43, 513)-Net over F64 — Constructive and digital
Digital (45, 88, 513)-net over F64, using
- t-expansion [i] based on digital (28, 88, 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
(45, 45+43, 516)-Net in Base 64 — Constructive
(45, 88, 516)-net in base 64, using
- base change [i] based on digital (23, 66, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 22, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 44, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 22, 258)-net over F256, using
- (u, u+v)-construction [i] based on
(45, 45+43, 1894)-Net over F64 — Digital
Digital (45, 88, 1894)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6488, 1894, F64, 2, 43) (dual of [(1894, 2), 3700, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6488, 2054, F64, 2, 43) (dual of [(2054, 2), 4020, 44]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6488, 4108, F64, 43) (dual of [4108, 4020, 44]-code), using
- construction X applied to C([0,21]) ⊂ C([0,19]) [i] based on
- linear OA(6485, 4097, F64, 43) (dual of [4097, 4012, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(6477, 4097, F64, 39) (dual of [4097, 4020, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to C([0,21]) ⊂ C([0,19]) [i] based on
- OOA 2-folding [i] based on linear OA(6488, 4108, F64, 43) (dual of [4108, 4020, 44]-code), using
- discarding factors / shortening the dual code based on linear OOA(6488, 2054, F64, 2, 43) (dual of [(2054, 2), 4020, 44]-NRT-code), using
(45, 45+43, 4186938)-Net in Base 64 — Upper bound on s
There is no (45, 88, 4186939)-net in base 64, because
- 1 times m-reduction [i] would yield (45, 87, 4186939)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 13 729649 236793 059675 428712 634311 912539 883262 693571 854893 099697 218125 742844 067001 563449 748332 564095 298280 110738 823806 487364 888369 067505 888380 056622 023551 287908 > 6487 [i]