Best Known (69−11, 69, s)-Nets in Base 8
(69−11, 69, 419440)-Net over F8 — Constructive and digital
Digital (58, 69, 419440)-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 (53, 64, 419431)-net over F8, using
- net defined by OOA [i] based on linear OOA(864, 419431, F8, 11, 11) (dual of [(419431, 11), 4613677, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(864, 2097156, F8, 11) (dual of [2097156, 2097092, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(864, 2097159, F8, 11) (dual of [2097159, 2097095, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(864, 2097152, F8, 11) (dual of [2097152, 2097088, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(857, 2097152, F8, 10) (dual of [2097152, 2097095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 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(864, 2097159, F8, 11) (dual of [2097159, 2097095, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(864, 2097156, F8, 11) (dual of [2097156, 2097092, 12]-code), using
- net defined by OOA [i] based on linear OOA(864, 419431, F8, 11, 11) (dual of [(419431, 11), 4613677, 12]-NRT-code), using
- digital (0, 5, 9)-net over F8, using
(69−11, 69, 2097179)-Net over F8 — Digital
Digital (58, 69, 2097179)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(869, 2097179, F8, 11) (dual of [2097179, 2097110, 12]-code), using
- construction XX applied to Ce(10) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- linear OA(864, 2097152, F8, 11) (dual of [2097152, 2097088, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(843, 2097152, F8, 7) (dual of [2097152, 2097109, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(836, 2097152, F8, 6) (dual of [2097152, 2097116, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(84, 26, F8, 3) (dual of [26, 22, 4]-code or 26-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
(69−11, 69, large)-Net in Base 8 — Upper bound on s
There is no (58, 69, large)-net in base 8, because
- 9 times m-reduction [i] would yield (58, 60, large)-net in base 8, but