Best Known (43, 43+17, s)-Nets in Base 8
(43, 43+17, 513)-Net over F8 — Constructive and digital
Digital (43, 60, 513)-net over F8, using
- 81 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(859, 513, F8, 17, 17) (dual of [(513, 17), 8662, 18]-NRT-code), using
(43, 43+17, 552)-Net in Base 8 — Constructive
(43, 60, 552)-net in base 8, using
- base change [i] based on digital (28, 45, 552)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- digital (17, 34, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 17, 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
- trace code for nets [i] based on digital (0, 17, 257)-net over F256, using
- digital (3, 11, 38)-net over F16, using
- (u, u+v)-construction [i] based on
(43, 43+17, 3264)-Net over F8 — Digital
Digital (43, 60, 3264)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(860, 3264, F8, 17) (dual of [3264, 3204, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(860, 4107, F8, 17) (dual of [4107, 4047, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(857, 4097, F8, 17) (dual of [4097, 4040, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(849, 4097, F8, 13) (dual of [4097, 4048, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(83, 10, F8, 3) (dual of [10, 7, 4]-code or 10-arc in PG(2,8) or 10-cap in PG(2,8)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(860, 4107, F8, 17) (dual of [4107, 4047, 18]-code), using
(43, 43+17, 2459687)-Net in Base 8 — Upper bound on s
There is no (43, 60, 2459688)-net in base 8, because
- 1 times m-reduction [i] would yield (43, 59, 2459688)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 191562 516523 832477 950714 876561 116437 992013 632003 952612 > 859 [i]