Best Known (36, 36+33, s)-Nets in Base 64
(36, 36+33, 513)-Net over F64 — Constructive and digital
Digital (36, 69, 513)-net over F64, using
- t-expansion [i] based on digital (28, 69, 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
(36, 36+33, 516)-Net in Base 64 — Constructive
(36, 69, 516)-net in base 64, using
- 641 times duplication [i] based on (35, 68, 516)-net in base 64, using
- base change [i] based on digital (18, 51, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 17, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 34, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 17, 258)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (18, 51, 516)-net over F256, using
(36, 36+33, 2052)-Net over F64 — Digital
Digital (36, 69, 2052)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6469, 2052, F64, 2, 33) (dual of [(2052, 2), 4035, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6469, 2055, F64, 2, 33) (dual of [(2055, 2), 4041, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6469, 4110, F64, 33) (dual of [4110, 4041, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(27) [i] based on
- linear OA(6465, 4096, F64, 33) (dual of [4096, 4031, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(6455, 4096, F64, 28) (dual of [4096, 4041, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(644, 14, F64, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(32) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(6469, 4110, F64, 33) (dual of [4110, 4041, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(6469, 2055, F64, 2, 33) (dual of [(2055, 2), 4041, 34]-NRT-code), using
(36, 36+33, 5122269)-Net in Base 64 — Upper bound on s
There is no (36, 69, 5122270)-net in base 64, because
- 1 times m-reduction [i] would yield (36, 68, 5122270)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 661 057421 983269 429995 347402 400637 329503 426480 677930 851225 301948 191083 138500 315992 648516 383335 454020 763240 410345 004048 044967 > 6468 [i]