Best Known (42, 42+17, s)-Nets in Base 8
(42, 42+17, 513)-Net over F8 — Constructive and digital
Digital (42, 59, 513)-net over F8, using
- net defined by OOA [i] based on linear OOA(859, 513, F8, 17, 17) (dual of [(513, 17), 8662, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(859, 4105, F8, 17) (dual of [4105, 4046, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(857, 4096, F8, 17) (dual of [4096, 4039, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(849, 4096, F8, 14) (dual of [4096, 4047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(16) ⊂ Ce(13) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(859, 4105, F8, 17) (dual of [4105, 4046, 18]-code), using
(42, 42+17, 542)-Net in Base 8 — Constructive
(42, 59, 542)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (5, 13, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- (29, 46, 514)-net in base 8, using
- trace code for nets [i] based on (6, 23, 257)-net in base 64, using
- 1 times m-reduction [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
- 1 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
- trace code for nets [i] based on (6, 23, 257)-net in base 64, using
- digital (5, 13, 28)-net over F8, using
(42, 42+17, 2840)-Net over F8 — Digital
Digital (42, 59, 2840)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(859, 2840, F8, 17) (dual of [2840, 2781, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(859, 4105, F8, 17) (dual of [4105, 4046, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(857, 4096, F8, 17) (dual of [4096, 4039, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(849, 4096, F8, 14) (dual of [4096, 4047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(859, 4105, F8, 17) (dual of [4105, 4046, 18]-code), using
(42, 42+17, 1896676)-Net in Base 8 — Upper bound on s
There is no (42, 59, 1896677)-net in base 8, because
- 1 times m-reduction [i] would yield (42, 58, 1896677)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 23945 247134 798944 093107 159426 227318 123361 359002 711148 > 858 [i]