Best Known (122−19, 122, s)-Nets in Base 8
(122−19, 122, 233026)-Net over F8 — Constructive and digital
Digital (103, 122, 233026)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 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]
- a shift-net [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, 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
- digital (0, 9, 9)-net over F8, using
(122−19, 122, 2097198)-Net over F8 — Digital
Digital (103, 122, 2097198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8122, 2097198, F8, 19) (dual of [2097198, 2097076, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8120, 2097194, F8, 19) (dual of [2097194, 2097074, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [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(878, 2097152, F8, 13) (dual of [2097152, 2097074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(87, 42, F8, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(8120, 2097196, F8, 18) (dual of [2097196, 2097076, 19]-code), using Gilbert–Varšamov bound and bm = 8120 > Vbs−1(k−1) = 192057 086439 886412 353824 913769 018049 553730 874692 028935 348223 522076 067357 745690 548194 639659 369378 502193 103420 [i]
- linear OA(80, 2, F8, 0) (dual of [2, 2, 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
- linear OA(8120, 2097194, F8, 19) (dual of [2097194, 2097074, 20]-code), using
- construction X with Varšamov bound [i] based on
(122−19, 122, large)-Net in Base 8 — Upper bound on s
There is no (103, 122, large)-net in base 8, because
- 17 times m-reduction [i] would yield (103, 105, large)-net in base 8, but