Best Known (35, 66, s)-Nets in Base 64
(35, 66, 513)-Net over F64 — Constructive and digital
Digital (35, 66, 513)-net over F64, using
- t-expansion [i] based on digital (28, 66, 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
(35, 66, 517)-Net in Base 64 — Constructive
(35, 66, 517)-net in base 64, using
- (u, u+v)-construction [i] based on
- (7, 22, 258)-net in base 64, using
- 2 times m-reduction [i] based on (7, 24, 258)-net in base 64, using
- base change [i] based on digital (1, 18, 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, 18, 258)-net over F256, using
- 2 times m-reduction [i] based on (7, 24, 258)-net in base 64, using
- (13, 44, 259)-net in base 64, using
- base change [i] based on digital (2, 33, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- base change [i] based on digital (2, 33, 259)-net over F256, using
- (7, 22, 258)-net in base 64, using
(35, 66, 2057)-Net over F64 — Digital
Digital (35, 66, 2057)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6466, 2057, F64, 31) (dual of [2057, 1991, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(6466, 4114, F64, 31) (dual of [4114, 4048, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [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(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(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6466, 4114, F64, 31) (dual of [4114, 4048, 32]-code), using
(35, 66, 6842353)-Net in Base 64 — Upper bound on s
There is no (35, 66, 6842354)-net in base 64, because
- 1 times m-reduction [i] would yield (35, 65, 6842354)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 2521 730156 177206 759395 817229 814505 889826 377144 448457 386142 479118 257752 145385 207969 401676 078343 430257 436173 315721 868264 > 6465 [i]