Best Known (31, 60, s)-Nets in Base 64
(31, 60, 513)-Net over F64 — Constructive and digital
Digital (31, 60, 513)-net over F64, using
- t-expansion [i] based on digital (28, 60, 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
(31, 60, 516)-Net in Base 64 — Constructive
(31, 60, 516)-net in base 64, using
- base change [i] based on digital (16, 45, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 15, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 30, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 15, 258)-net over F256, using
- (u, u+v)-construction [i] based on
(31, 60, 1779)-Net over F64 — Digital
Digital (31, 60, 1779)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6460, 1779, F64, 2, 29) (dual of [(1779, 2), 3498, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6460, 2054, F64, 2, 29) (dual of [(2054, 2), 4048, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6460, 4108, F64, 29) (dual of [4108, 4048, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- 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(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(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([0,14]) ⊂ C([0,12]) [i] based on
- OOA 2-folding [i] based on linear OA(6460, 4108, F64, 29) (dual of [4108, 4048, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(6460, 2054, F64, 2, 29) (dual of [(2054, 2), 4048, 30]-NRT-code), using
(31, 60, 3925318)-Net in Base 64 — Upper bound on s
There is no (31, 60, 3925319)-net in base 64, because
- 1 times m-reduction [i] would yield (31, 59, 3925319)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 36696 035132 355454 184573 597440 953871 884660 961762 824235 286943 727542 458798 086931 981137 828683 833066 110819 813000 > 6459 [i]