Best Known (33, 33+31, s)-Nets in Base 64
(33, 33+31, 513)-Net over F64 — Constructive and digital
Digital (33, 64, 513)-net over F64, using
- t-expansion [i] based on digital (28, 64, 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
(33, 33+31, 516)-Net in Base 64 — Constructive
(33, 64, 516)-net in base 64, using
- base change [i] based on digital (17, 48, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 16, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 32, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 16, 258)-net over F256, using
- (u, u+v)-construction [i] based on
(33, 33+31, 1777)-Net over F64 — Digital
Digital (33, 64, 1777)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6464, 1777, F64, 2, 31) (dual of [(1777, 2), 3490, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6464, 2054, F64, 2, 31) (dual of [(2054, 2), 4044, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6464, 4108, F64, 31) (dual of [4108, 4044, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(6461, 4097, F64, 31) (dual of [4097, 4036, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(6453, 4097, F64, 27) (dual of [4097, 4044, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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([0,15]) ⊂ C([0,13]) [i] based on
- OOA 2-folding [i] based on linear OA(6464, 4108, F64, 31) (dual of [4108, 4044, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(6464, 2054, F64, 2, 31) (dual of [(2054, 2), 4044, 32]-NRT-code), using
(33, 33+31, 3929897)-Net in Base 64 — Upper bound on s
There is no (33, 64, 3929898)-net in base 64, because
- 1 times m-reduction [i] would yield (33, 63, 3929898)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 615658 395125 316149 702864 105248 546168 011217 417954 884211 139477 093394 652045 054727 271851 237568 008414 320675 267674 828976 > 6463 [i]