Best Known (128−19, 128, s)-Nets in Base 8
(128−19, 128, 233046)-Net over F8 — Constructive and digital
Digital (109, 128, 233046)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (5, 14, 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
- digital (95, 114, 233018)-net over F8, using
- net defined by OOA [i] based on linear OOA(8114, 233018, F8, 19, 19) (dual of [(233018, 19), 4427228, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8114, 2097163, F8, 19) (dual of [2097163, 2097049, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8114, 2097168, F8, 19) (dual of [2097168, 2097054, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(8113, 2097153, F8, 19) (dual of [2097153, 2097040, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(899, 2097153, F8, 17) (dual of [2097153, 2097054, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(81, 15, F8, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8114, 2097168, F8, 19) (dual of [2097168, 2097054, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8114, 2097163, F8, 19) (dual of [2097163, 2097049, 20]-code), using
- net defined by OOA [i] based on linear OOA(8114, 233018, F8, 19, 19) (dual of [(233018, 19), 4427228, 20]-NRT-code), using
- digital (5, 14, 28)-net over F8, using
(128−19, 128, 233050)-Net in Base 8 — Constructive
(109, 128, 233050)-net in base 8, using
- (u, u+v)-construction [i] based on
- (6, 15, 33)-net in base 8, using
- 1 times m-reduction [i] based on (6, 16, 33)-net in base 8, using
- base change [i] based on digital (2, 12, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- base change [i] based on digital (2, 12, 33)-net over F16, using
- 1 times m-reduction [i] based on (6, 16, 33)-net in base 8, using
- digital (94, 113, 233017)-net over F8, using
- net defined by OOA [i] based on linear OOA(8113, 233017, F8, 19, 19) (dual of [(233017, 19), 4427210, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8113, 2097154, F8, 19) (dual of [2097154, 2097041, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8113, 2097159, F8, 19) (dual of [2097159, 2097046, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(8113, 2097152, F8, 19) (dual of [2097152, 2097039, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(8106, 2097152, F8, 18) (dual of [2097152, 2097046, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(8113, 2097159, F8, 19) (dual of [2097159, 2097046, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8113, 2097154, F8, 19) (dual of [2097154, 2097041, 20]-code), using
- net defined by OOA [i] based on linear OOA(8113, 233017, F8, 19, 19) (dual of [(233017, 19), 4427210, 20]-NRT-code), using
- (6, 15, 33)-net in base 8, using
(128−19, 128, 2851062)-Net over F8 — Digital
Digital (109, 128, 2851062)-net over F8, using
(128−19, 128, large)-Net in Base 8 — Upper bound on s
There is no (109, 128, large)-net in base 8, because
- 17 times m-reduction [i] would yield (109, 111, large)-net in base 8, but