Best Known (40, 40+11, s)-Nets in Base 8
(40, 40+11, 6563)-Net over F8 — Constructive and digital
Digital (40, 51, 6563)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (35, 46, 6554)-net over F8, using
- net defined by OOA [i] based on linear OOA(846, 6554, F8, 11, 11) (dual of [(6554, 11), 72048, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(846, 32771, F8, 11) (dual of [32771, 32725, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(846, 32773, F8, 11) (dual of [32773, 32727, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(846, 32768, F8, 11) (dual of [32768, 32722, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(841, 32768, F8, 10) (dual of [32768, 32727, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(80, 5, F8, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(846, 32773, F8, 11) (dual of [32773, 32727, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(846, 32771, F8, 11) (dual of [32771, 32725, 12]-code), using
- net defined by OOA [i] based on linear OOA(846, 6554, F8, 11, 11) (dual of [(6554, 11), 72048, 12]-NRT-code), using
- digital (0, 5, 9)-net over F8, using
(40, 40+11, 32789)-Net over F8 — Digital
Digital (40, 51, 32789)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(851, 32789, F8, 11) (dual of [32789, 32738, 12]-code), using
- construction XX applied to Ce(10) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- linear OA(846, 32768, F8, 11) (dual of [32768, 32722, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(831, 32768, F8, 7) (dual of [32768, 32737, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(826, 32768, F8, 6) (dual of [32768, 32742, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(84, 20, F8, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,8)), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(10) ⊂ Ce(6) ⊂ Ce(5) [i] based on
(40, 40+11, large)-Net in Base 8 — Upper bound on s
There is no (40, 51, large)-net in base 8, because
- 9 times m-reduction [i] would yield (40, 42, large)-net in base 8, but