Best Known (74−36, 74, s)-Nets in Base 64
(74−36, 74, 513)-Net over F64 — Constructive and digital
Digital (38, 74, 513)-net over F64, using
- t-expansion [i] based on digital (28, 74, 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
(74−36, 74, 515)-Net in Base 64 — Constructive
(38, 74, 515)-net in base 64, using
- (u, u+v)-construction [i] based on
- (6, 24, 257)-net in base 64, using
- base change [i] based on digital (0, 18, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 18, 257)-net over F256, using
- (14, 50, 258)-net in base 64, using
- 2 times m-reduction [i] based on (14, 52, 258)-net in base 64, using
- base change [i] based on digital (1, 39, 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, 39, 258)-net over F256, using
- 2 times m-reduction [i] based on (14, 52, 258)-net in base 64, using
- (6, 24, 257)-net in base 64, using
(74−36, 74, 1807)-Net over F64 — Digital
Digital (38, 74, 1807)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6474, 1807, F64, 2, 36) (dual of [(1807, 2), 3540, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6474, 2053, F64, 2, 36) (dual of [(2053, 2), 4032, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6474, 4106, F64, 36) (dual of [4106, 4032, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(6474, 4107, F64, 36) (dual of [4107, 4033, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(31) [i] based on
- linear OA(6471, 4096, F64, 36) (dual of [4096, 4025, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(6463, 4096, F64, 32) (dual of [4096, 4033, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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 Ce(35) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(6474, 4107, F64, 36) (dual of [4107, 4033, 37]-code), using
- OOA 2-folding [i] based on linear OA(6474, 4106, F64, 36) (dual of [4106, 4032, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(6474, 2053, F64, 2, 36) (dual of [(2053, 2), 4032, 37]-NRT-code), using
(74−36, 74, 3192970)-Net in Base 64 — Upper bound on s
There is no (38, 74, 3192971)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 45 427612 605019 727827 058966 148882 344059 756876 690212 200166 160867 595361 529888 773938 748243 304557 980756 559626 586619 949363 540388 812711 132104 > 6474 [i]