Best Known (136−20, 136, s)-Nets in Base 8
(136−20, 136, 209744)-Net over F8 — Constructive and digital
Digital (116, 136, 209744)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (5, 15, 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 (101, 121, 209716)-net over F8, using
- net defined by OOA [i] based on linear OOA(8121, 209716, F8, 20, 20) (dual of [(209716, 20), 4194199, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8121, 2097160, F8, 20) (dual of [2097160, 2097039, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8121, 2097167, F8, 20) (dual of [2097167, 2097046, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(8120, 2097152, F8, 20) (dual of [2097152, 2097032, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(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 Ce(19) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(8121, 2097167, F8, 20) (dual of [2097167, 2097046, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(8121, 2097160, F8, 20) (dual of [2097160, 2097039, 21]-code), using
- net defined by OOA [i] based on linear OOA(8121, 209716, F8, 20, 20) (dual of [(209716, 20), 4194199, 21]-NRT-code), using
- digital (5, 15, 28)-net over F8, using
(136−20, 136, 419431)-Net in Base 8 — Constructive
(116, 136, 419431)-net in base 8, using
- net defined by OOA [i] based on OOA(8136, 419431, S8, 20, 20), using
- OA 10-folding and stacking [i] based on OA(8136, 4194310, S8, 20), using
- trace code [i] based on OA(6468, 2097155, S64, 20), using
- discarding parts of the base [i] based on linear OA(12858, 2097155, F128, 20) (dual of [2097155, 2097097, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- discarding parts of the base [i] based on linear OA(12858, 2097155, F128, 20) (dual of [2097155, 2097097, 21]-code), using
- trace code [i] based on OA(6468, 2097155, S64, 20), using
- OA 10-folding and stacking [i] based on OA(8136, 4194310, S8, 20), using
(136−20, 136, 3298692)-Net over F8 — Digital
Digital (116, 136, 3298692)-net over F8, using
(136−20, 136, large)-Net in Base 8 — Upper bound on s
There is no (116, 136, large)-net in base 8, because
- 18 times m-reduction [i] would yield (116, 118, large)-net in base 8, but