Best Known (64−18, 64, s)-Nets in Base 8
(64−18, 64, 456)-Net over F8 — Constructive and digital
Digital (46, 64, 456)-net over F8, using
- 81 times duplication [i] based on digital (45, 63, 456)-net over F8, using
- net defined by OOA [i] based on linear OOA(863, 456, F8, 18, 18) (dual of [(456, 18), 8145, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(863, 4104, F8, 18) (dual of [4104, 4041, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(863, 4105, F8, 18) (dual of [4105, 4042, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(861, 4096, F8, 18) (dual of [4096, 4035, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(853, 4096, F8, 15) (dual of [4096, 4043, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(82, 9, F8, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,8)), using
- extended Reed–Solomon code RSe(7,8) [i]
- Hamming code H(2,8) [i]
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(863, 4105, F8, 18) (dual of [4105, 4042, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(863, 4104, F8, 18) (dual of [4104, 4041, 19]-code), using
- net defined by OOA [i] based on linear OOA(863, 456, F8, 18, 18) (dual of [(456, 18), 8145, 19]-NRT-code), using
(64−18, 64, 576)-Net in Base 8 — Constructive
(46, 64, 576)-net in base 8, using
- trace code for nets [i] based on (14, 32, 288)-net in base 64, using
- 3 times m-reduction [i] based on (14, 35, 288)-net in base 64, using
- base change [i] based on digital (9, 30, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 30, 288)-net over F128, using
- 3 times m-reduction [i] based on (14, 35, 288)-net in base 64, using
(64−18, 64, 3485)-Net over F8 — Digital
Digital (46, 64, 3485)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(864, 3485, F8, 18) (dual of [3485, 3421, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(864, 4107, F8, 18) (dual of [4107, 4043, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(861, 4096, F8, 18) (dual of [4096, 4035, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(853, 4096, F8, 15) (dual of [4096, 4043, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(83, 11, F8, 2) (dual of [11, 8, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(864, 4107, F8, 18) (dual of [4107, 4043, 19]-code), using
(64−18, 64, 1565399)-Net in Base 8 — Upper bound on s
There is no (46, 64, 1565400)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 6277 120857 409275 648785 898545 627460 691750 640127 730650 868786 > 864 [i]